example hypothesis tests

Statistics Made Easy

Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses.

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1.

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results.

Interpret the results of the hypothesis test in the context of the question being asked.

The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta , denoted as β.

One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related: What is a Directional Hypothesis?

Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

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Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as $H_{0}$. Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as $H_{1}$ or $H_{a}$. For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, $\alpha$ or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$. $\overline{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation and n is the size of the sample.
t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$. s is the sample standard deviation.
$\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}$. $O_{i}$ is the observed value and $E_{i}$ is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

One sample: z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.
Two samples: z = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$.

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

One sample: t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$.
Two samples: t = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$.

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

$H_{0}$: The population parameter is ≤ some value

$H_{1}$: The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

$H_{0}$: The population parameter is ≥ some value

$H_{1}$: The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

$H_{0}$: the population parameter = some value

$H_{1}$: the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
Step 2: Set up the alternative hypothesis.
Step 3: Choose the correct significance level, $\alpha$, and find the critical value.
Step 4: Calculate the correct test statistic (z, t or $\chi$) and p-value.
Step 5: Compare the test statistic with the critical value or compare the p-value with $\alpha$ to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as $H_{0}$: $\mu$ = 100.

Step 2: The alternative hypothesis is given by $H_{1}$: $\mu$ > 100.

Step 3: As this is a one-tailed test, $\alpha$ = 100% - 95% = 5%. This can be used to determine the critical value.

1 - $\alpha$ = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.

$\mu$ = 100, $\overline{x}$ = 112.5, n = 30, $\sigma$ = 15

z = $\frac{112.5-100}{\frac{15}{\sqrt{30}}}$ = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Probability and Statistics
Data Handling

Important Notes on Hypothesis Testing

Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
It involves the setting up of a null hypothesis and an alternate hypothesis.
There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. $H_{0}$: $\mu$ = 90, $H_{1}$: $\mu$ > 90 $\overline{x}$ = 110, $\mu$ = 90, n = 5, s = 18. $\alpha$ = 0.05 Using the t-distribution table, the critical value is 2.132 t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$ t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. $H_{0}$: $\mu$ = 80, $H_{1}$: $\mu$ ≠ 80 $\overline{x}$ = 88, $\mu$ = 80, n = 36, $\sigma$ = 10. $\alpha$ = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ z = $\frac{88-80}{\frac{10}{\sqrt{36}}}$ = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. $H_{0}$: $\mu$ = 90, $H_{1}$: $\mu$ < 90 $\overline{x}$ = 110, $\mu$ = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = $\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$ t = $\frac{82-90}{\frac{18}{\sqrt{6}}}$ t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = $\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ and for two samples is z = $\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$.

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide $\alpha$ by 2. This is done as there are two rejection regions in the curve.

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6a.2 - steps for hypothesis tests, the logic of hypothesis testing section .

A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.

How do we decide whether to reject the null hypothesis?

If the sample data are consistent with the null hypothesis, then we do not reject it.
If the sample data are inconsistent with the null hypothesis, but consistent with the alternative, then we reject the null hypothesis and conclude that the alternative hypothesis is true.

Six Steps for Hypothesis Tests Section

In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.

Set up the hypotheses and check conditions : Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as $H_0 $, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is evidence to suggest otherwise. The second hypothesis is called the alternative, or research hypothesis, notated as $H_a $. The alternative hypothesis is a statement of a range of alternative values in which the parameter may fall. One must also check that any conditions (assumptions) needed to run the test have been satisfied e.g. normality of data, independence, and number of success and failure outcomes.
Decide on the significance level, $\alpha $: This value is used as a probability cutoff for making decisions about the null hypothesis. This alpha value represents the probability we are willing to place on our test for making an incorrect decision in regards to rejecting the null hypothesis. The most common $\alpha $ value is 0.05 or 5%. Other popular choices are 0.01 (1%) and 0.1 (10%).
Calculate the test statistic: Gather sample data and calculate a test statistic where the sample statistic is compared to the parameter value. The test statistic is calculated under the assumption the null hypothesis is true and incorporates a measure of standard error and assumptions (conditions) related to the sampling distribution.
Calculate probability value (p-value), or find the rejection region: A p-value is found by using the test statistic to calculate the probability of the sample data producing such a test statistic or one more extreme. The rejection region is found by using alpha to find a critical value; the rejection region is the area that is more extreme than the critical value. We discuss the p-value and rejection region in more detail in the next section.
Make a decision about the null hypothesis: In this step, we decide to either reject the null hypothesis or decide to fail to reject the null hypothesis. Notice we do not make a decision where we will accept the null hypothesis.
State an overall conclusion : Once we have found the p-value or rejection region, and made a statistical decision about the null hypothesis (i.e. we will reject the null or fail to reject the null), we then want to summarize our results into an overall conclusion for our test.

We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.

Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.

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9.E: Hypothesis Testing with One Sample (Exercises)

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These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

9.1: Introduction

9.2: null and alternative hypotheses.

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, $H_{0}$, and the alternative hypothesis. $H_{a}$, in terms of the appropriate parameter $(\mu \text{or} p)$.

The mean number of years Americans work before retiring is 34.
At most 60% of Americans vote in presidential elections.
The mean starting salary for San Jose State University graduates is at least $100,000 per year.
Twenty-nine percent of high school seniors get drunk each month.
Fewer than 5% of adults ride the bus to work in Los Angeles.
The mean number of cars a person owns in her lifetime is not more than ten.
About half of Americans prefer to live away from cities, given the choice.
Europeans have a mean paid vacation each year of six weeks.
The chance of developing breast cancer is under 11% for women.
Private universities' mean tuition cost is more than $20,000 per year.
$H_{0}: \mu = 34; H_{a}: \mu \neq 34$
$H_{0}: p \leq 0.60; H_{a}: p > 0.60$
$H_{0}: \mu \geq 100,000; H_{a}: \mu < 100,000$
$H_{0}: p = 0.29; H_{a}: p \neq 0.29$
$H_{0}: p = 0.05; H_{a}: p < 0.05$
$H_{0}: \mu \leq 10; H_{a}: \mu > 10$
$H_{0}: p = 0.50; H_{a}: p \neq 0.50$
$H_{0}: \mu = 6; H_{a}: \mu \neq 6$
$H_{0}: p ≥ 0.11; H_{a}: p < 0.11$
$H_{0}: \mu \leq 20,000; H_{a}: \mu > 20,000$

Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:

$p < 0.30$
$p \leq 0.30$
$p \geq 0.30$
$p > 0.30$

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:

$p = 0.20$
$p > 0.20$
$p < 0.20$
$p \leq 0.20$

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:

$H_{0}: \bar{x} = 4.5, H_{a}: \bar{x} > 4.5$
$H_{0}: \mu \geq 4.5, H_{a}: \mu < 4.5$
$H_{0}: \mu = 4.75, H_{a}: \mu > 4.75$
$H_{0}: \mu = 4.5, H_{a}: \mu > 4.5$

9.3: Outcomes and the Type I and Type II Errors

State the Type I and Type II errors in complete sentences given the following statements.

The mean number of cars a person owns in his or her lifetime is not more than ten.
Private universities mean tuition cost is more than $20,000 per year.
Type I error: We conclude that the mean is not 34 years, when it really is 34 years. Type II error: We conclude that the mean is 34 years, when in fact it really is not 34 years.
Type I error: We conclude that more than 60% of Americans vote in presidential elections, when the actual percentage is at most 60%.Type II error: We conclude that at most 60% of Americans vote in presidential elections when, in fact, more than 60% do.
Type I error: We conclude that the mean starting salary is less than $100,000, when it really is at least $100,000. Type II error: We conclude that the mean starting salary is at least $100,000 when, in fact, it is less than $100,000.
Type I error: We conclude that the proportion of high school seniors who get drunk each month is not 29%, when it really is 29%. Type II error: We conclude that the proportion of high school seniors who get drunk each month is 29% when, in fact, it is not 29%.
Type I error: We conclude that fewer than 5% of adults ride the bus to work in Los Angeles, when the percentage that do is really 5% or more. Type II error: We conclude that 5% or more adults ride the bus to work in Los Angeles when, in fact, fewer that 5% do.
Type I error: We conclude that the mean number of cars a person owns in his or her lifetime is more than 10, when in reality it is not more than 10. Type II error: We conclude that the mean number of cars a person owns in his or her lifetime is not more than 10 when, in fact, it is more than 10.
Type I error: We conclude that the proportion of Americans who prefer to live away from cities is not about half, though the actual proportion is about half. Type II error: We conclude that the proportion of Americans who prefer to live away from cities is half when, in fact, it is not half.
Type I error: We conclude that the duration of paid vacations each year for Europeans is not six weeks, when in fact it is six weeks. Type II error: We conclude that the duration of paid vacations each year for Europeans is six weeks when, in fact, it is not.
Type I error: We conclude that the proportion is less than 11%, when it is really at least 11%. Type II error: We conclude that the proportion of women who develop breast cancer is at least 11%, when in fact it is less than 11%.
Type I error: We conclude that the average tuition cost at private universities is more than $20,000, though in reality it is at most $20,000. Type II error: We conclude that the average tuition cost at private universities is at most $20,000 when, in fact, it is more than $20,000.

For statements a-j in Exercise 9.109 , answer the following in complete sentences.

State a consequence of committing a Type I error.
State a consequence of committing a Type II error.

When a new drug is created, the pharmaceutical company must subject it to testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug. Suppose the null hypothesis is “the drug is unsafe.” What is the Type II Error?

To conclude the drug is safe when in, fact, it is unsafe.
Not to conclude the drug is safe when, in fact, it is safe.
To conclude the drug is safe when, in fact, it is safe.
Not to conclude the drug is unsafe when, in fact, it is unsafe.

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 of them attended the midnight showing. The Type I error is to conclude that the percent of EVC students who attended is ________.

at least 20%, when in fact, it is less than 20%.
20%, when in fact, it is 20%.
less than 20%, when in fact, it is at least 20%.
less than 20%, when in fact, it is less than 20%.

It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average?

The Type II error is not to reject that the mean number of hours of sleep LTCC students get per night is at least seven when, in fact, the mean number of hours

is more than seven hours.
is at most seven hours.
is at least seven hours.
is less than seven hours.

to conclude that the current mean hours per week is higher than 4.5, when in fact, it is higher
to conclude that the current mean hours per week is higher than 4.5, when in fact, it is the same
to conclude that the mean hours per week currently is 4.5, when in fact, it is higher
to conclude that the mean hours per week currently is no higher than 4.5, when in fact, it is not higher

9.4: Distribution Needed for Hypothesis Testing

$N\left(7.24, \frac{1.93}{\sqrt{22}}\right)$
$N\left(7.24, 1.93\right)$

9.5: Rare Events, the Sample, Decision and Conclusion

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. Conduct a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population.

Is this a test of one mean or proportion?
State the null and alternative hypotheses. $H_{0}$ : ____________________ $H_{a}$ : ____________________
Is this a right-tailed, left-tailed, or two-tailed test?
What symbol represents the random variable for this test?
In words, define the random variable for this test.
$x =$ ________________
$n =$ ________________
$p′ =$ _____________
Calculate $\sigma_{x} =$ __________. Show the formula set-up.
State the distribution to use for the hypothesis test.
Find the $p\text{-value}$.
Reason for the decision:
Conclusion (write out in a complete sentence):

9.6: Additional Information and Full Hypothesis Test Examples

For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in [link] . Please feel free to make copies of the solution sheets. For the online version of the book, it is suggested that you copy the .doc or the .pdf files.

If you are using a Student's $t$ - distribution for one of the following homework problems, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, however.)

A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles. Using $\alpha = 0.05$, is the data highly inconsistent with the claim?

$H_{0}: \mu \geq 50,000$
$H_{a}: \mu < 50,000$
Let $\bar{X} =$ the average lifespan of a brand of tires.
normal distribution
$z = -2.315$
$p\text{-value} = 0.0103$
Check student’s solution.
alpha: 0.05
Decision: Reject the null hypothesis.
Reason for decision: The $p\text{-value}$ is less than 0.05.
Conclusion: There is sufficient evidence to conclude that the mean lifespan of the tires is less than 50,000 miles.
$(43,537, 49,463)$

From generation to generation, the mean age when smokers first start to smoke varies. However, the standard deviation of that age remains constant of around 2.1 years. A survey of 40 smokers of this generation was done to see if the mean starting age is at least 19. The sample mean was 18.1 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level?

The cost of a daily newspaper varies from city to city. However, the variation among prices remains steady with a standard deviation of 20¢. A study was done to test the claim that the mean cost of a daily newspaper is $1.00. Twelve costs yield a mean cost of 95¢ with a standard deviation of 18¢. Do the data support the claim at the 1% level?

$H_{0}: \mu = $1.00$
$H_{a}: \mu \neq $1.00$
Let $\bar{X} =$ the average cost of a daily newspaper.
$z = –0.866$
$p\text{-value} = 0.3865$
$\alpha: 0.01$
Decision: Do not reject the null hypothesis.
Reason for decision: The $p\text{-value}$ is greater than 0.01.
Conclusion: There is sufficient evidence to support the claim that the mean cost of daily papers is $1. The mean cost could be $1.
$($0.84, $1.06)$

An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% level?

The mean number of sick days an employee takes per year is believed to be about ten. Members of a personnel department do not believe this figure. They randomly survey eight employees. The number of sick days they took for the past year are as follows: 12; 4; 15; 3; 11; 8; 6; 8. Let $x =$ the number of sick days they took for the past year. Should the personnel team believe that the mean number is ten?

$H_{0}: \mu = 10$
$H_{a}: \mu \neq 10$
Let $\bar{X}$ the mean number of sick days an employee takes per year.
Student’s t -distribution
$t = –1.12$
$p\text{-value} = 0.300$
$\alpha: 0.05$
Reason for decision: The $p\text{-value}$ is greater than 0.05.
Conclusion: At the 5% significance level, there is insufficient evidence to conclude that the mean number of sick days is not ten.
$(4.9443, 11.806)$

In 1955, Life Magazine reported that the 25 year-old mother of three worked, on average, an 80 hour week. Recently, many groups have been studying whether or not the women's movement has, in fact, resulted in an increase in the average work week for women (combining employment and at-home work). Suppose a study was done to determine if the mean work week has increased. 81 women were surveyed with the following results. The sample mean was 83; the sample standard deviation was ten. Does it appear that the mean work week has increased for women at the 5% level?

Your statistics instructor claims that 60 percent of the students who take her Elementary Statistics class go through life feeling more enriched. For some reason that she can't quite figure out, most people don't believe her. You decide to check this out on your own. You randomly survey 64 of her past Elementary Statistics students and find that 34 feel more enriched as a result of her class. Now, what do you think?

$H_{0}: p \geq 0.6$
$H_{a}: p < 0.6$
Let $P′ =$ the proportion of students who feel more enriched as a result of taking Elementary Statistics.
normal for a single proportion
$p\text{-value} = 0.1308$
Conclusion: There is insufficient evidence to conclude that less than 60 percent of her students feel more enriched.

The “plus-4s” confidence interval is $(0.411, 0.648)$

A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.

Refer to Exercise 9.119 . Conduct a hypothesis test to see if your decision and conclusion would change if your belief were that the brown trout’s mean I.Q. is not four.

$H_{0}: \mu = 4$
$H_{a}: \mu \neq 4$
Let $\bar{X}$ the average I.Q. of a set of brown trout.
two-tailed Student's t-test
$t = 1.95$
$p\text{-value} = 0.076$
Reason for decision: The $p\text{-value}$ is greater than 0.05
Conclusion: There is insufficient evidence to conclude that the average IQ of brown trout is not four.
$(3.8865,5.9468)$

According to an article in Newsweek , the natural ratio of girls to boys is 100:105. In China, the birth ratio is 100: 114 (46.7% girls). Suppose you don’t believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 60 girls and 90 boys born of the 150. Based on your study, do you believe that the percent of girls born in China is 46.7?

A poll done for Newsweek found that 13% of Americans have seen or sensed the presence of an angel. A contingent doubts that the percent is really that high. It conducts its own survey. Out of 76 Americans surveyed, only two had seen or sensed the presence of an angel. As a result of the contingent’s survey, would you agree with the Newsweek poll? In complete sentences, also give three reasons why the two polls might give different results.

$H_{a}: p < 0.13$
Let $P′ =$ the proportion of Americans who have seen or sensed angels
–2.688
$p\text{-value} = 0.0036$
Reason for decision: The $p\text{-value}$e is less than 0.05.
Conclusion: There is sufficient evidence to conclude that the percentage of Americans who have seen or sensed an angel is less than 13%.

The“plus-4s” confidence interval is (0.0022, 0.0978)

The mean work week for engineers in a start-up company is believed to be about 60 hours. A newly hired engineer hopes that it’s shorter. She asks ten engineering friends in start-ups for the lengths of their mean work weeks. Based on the results that follow, should she count on the mean work week to be shorter than 60 hours?

Data (length of mean work week): 70; 45; 55; 60; 65; 55; 55; 60; 50; 55.

Use the “Lap time” data for Lap 4 (see [link] ) to test the claim that Terri finishes Lap 4, on average, in less than 129 seconds. Use all twenty races given.

$H_{0}: \mu \geq 129$
$H_{a}: \mu < 129$
Let $\bar{X} =$ the average time in seconds that Terri finishes Lap 4.
Student's t -distribution
$t = 1.209$
Conclusion: There is insufficient evidence to conclude that Terri’s mean lap time is less than 129 seconds.
$(128.63, 130.37)$

Use the “Initial Public Offering” data (see [link] ) to test the claim that the mean offer price was $18 per share. Do not use all the data. Use your random number generator to randomly survey 15 prices.

The following questions were written by past students. They are excellent problems!

"Asian Family Reunion," by Chau Nguyen

Every two years it comes around.

We all get together from different towns.

In my honest opinion,

It's not a typical family reunion.

Not forty, or fifty, or sixty,

But how about seventy companions!

The kids would play, scream, and shout

One minute they're happy, another they'll pout.

The teenagers would look, stare, and compare

From how they look to what they wear.

The men would chat about their business

That they make more, but never less.

Money is always their subject

And there's always talk of more new projects.

The women get tired from all of the chats

They head to the kitchen to set out the mats.

Some would sit and some would stand

Eating and talking with plates in their hands.

Then come the games and the songs

And suddenly, everyone gets along!

With all that laughter, it's sad to say

That it always ends in the same old way.

They hug and kiss and say "good-bye"

And then they all begin to cry!

I say that 60 percent shed their tears

But my mom counted 35 people this year.

She said that boys and men will always have their pride,

So we won't ever see them cry.

I myself don't think she's correct,

So could you please try this problem to see if you object?

$H_{0}: p = 0.60$
$H_{a}: p < 0.60$
Let $P′ =$ the proportion of family members who shed tears at a reunion.
–1.71
Reason for decision: $p\text{-value} < \alpha$
Conclusion: At the 5% significance level, there is sufficient evidence to conclude that the proportion of family members who shed tears at a reunion is less than 0.60. However, the test is weak because the $p\text{-value}$ and alpha are quite close, so other tests should be done.
We are 95% confident that between 38.29% and 61.71% of family members will shed tears at a family reunion. $(0.3829, 0.6171)$. The“plus-4s” confidence interval (see chapter 8) is $(0.3861, 0.6139)$

Note that here the “large-sample” $1 - \text{PropZTest}$ provides the approximate $p\text{-value}$ of 0.0438. Whenever a $p\text{-value}$ based on a normal approximation is close to the level of significance, the exact $p\text{-value}$ based on binomial probabilities should be calculated whenever possible. This is beyond the scope of this course.

"The Problem with Angels," by Cyndy Dowling

Although this problem is wholly mine,

The catalyst came from the magazine, Time.

On the magazine cover I did find

The realm of angels tickling my mind.

Inside, 69% I found to be

In angels, Americans do believe.

Then, it was time to rise to the task,

Ninety-five high school and college students I did ask.

Viewing all as one group,

Random sampling to get the scoop.

So, I asked each to be true,

"Do you believe in angels?" Tell me, do!

Hypothesizing at the start,

Totally believing in my heart

That the proportion who said yes

Would be equal on this test.

Lo and behold, seventy-three did arrive,

Out of the sample of ninety-five.

Now your job has just begun,

Solve this problem and have some fun.

"Blowing Bubbles," by Sondra Prull

Studying stats just made me tense,

I had to find some sane defense.

Some light and lifting simple play

To float my math anxiety away.

Blowing bubbles lifts me high

Takes my troubles to the sky.

POIK! They're gone, with all my stress

Bubble therapy is the best.

The label said each time I blew

The average number of bubbles would be at least 22.

I blew and blew and this I found

From 64 blows, they all are round!

But the number of bubbles in 64 blows

Varied widely, this I know.

20 per blow became the mean

They deviated by 6, and not 16.

From counting bubbles, I sure did relax

But now I give to you your task.

Was 22 a reasonable guess?

Find the answer and pass this test!

$H_{0}: \mu \geq 22$
$H_{a}: \mu < 22$
Let $\bar{X} =$ the mean number of bubbles per blow.
–2.667
$p\text{-value} = 0.00486$
Conclusion: There is sufficient evidence to conclude that the mean number of bubbles per blow is less than 22.
$(18.501, 21.499)$

"Dalmatian Darnation," by Kathy Sparling

A greedy dog breeder named Spreckles

Bred puppies with numerous freckles

The Dalmatians he sought

Possessed spot upon spot

The more spots, he thought, the more shekels.

His competitors did not agree

That freckles would increase the fee.

They said, “Spots are quite nice

But they don't affect price;

One should breed for improved pedigree.”

The breeders decided to prove

This strategy was a wrong move.

Breeding only for spots

Would wreak havoc, they thought.

His theory they want to disprove.

They proposed a contest to Spreckles

Comparing dog prices to freckles.

In records they looked up

One hundred one pups:

Dalmatians that fetched the most shekels.

They asked Mr. Spreckles to name

An average spot count he'd claim

To bring in big bucks.

Said Spreckles, “Well, shucks,

It's for one hundred one that I aim.”

Said an amateur statistician

Who wanted to help with this mission.

“Twenty-one for the sample

Standard deviation's ample:

They examined one hundred and one

Dalmatians that fetched a good sum.

They counted each spot,

Mark, freckle and dot

And tallied up every one.

Instead of one hundred one spots

They averaged ninety six dots

Can they muzzle Spreckles’

Obsession with freckles

Based on all the dog data they've got?

"Macaroni and Cheese, please!!" by Nedda Misherghi and Rachelle Hall

As a poor starving student I don't have much money to spend for even the bare necessities. So my favorite and main staple food is macaroni and cheese. It's high in taste and low in cost and nutritional value.

One day, as I sat down to determine the meaning of life, I got a serious craving for this, oh, so important, food of my life. So I went down the street to Greatway to get a box of macaroni and cheese, but it was SO expensive! $2.02 !!! Can you believe it? It made me stop and think. The world is changing fast. I had thought that the mean cost of a box (the normal size, not some super-gigantic-family-value-pack) was at most $1, but now I wasn't so sure. However, I was determined to find out. I went to 53 of the closest grocery stores and surveyed the prices of macaroni and cheese. Here are the data I wrote in my notebook:

Price per box of Mac and Cheese:

5 stores @ $2.02
15 stores @ $0.25
3 stores @ $1.29
6 stores @ $0.35
4 stores @ $2.27
7 stores @ $1.50
5 stores @ $1.89
8 stores @ 0.75.

I could see that the cost varied but I had to sit down to figure out whether or not I was right. If it does turn out that this mouth-watering dish is at most $1, then I'll throw a big cheesy party in our next statistics lab, with enough macaroni and cheese for just me. (After all, as a poor starving student I can't be expected to feed our class of animals!)

$H_{0}: \mu \leq 1$
$H_{a}: \mu > 1$
Let $\bar{X} =$ the mean cost in dollars of macaroni and cheese in a certain town.
Student's $t$-distribution
$t = 0.340$
$p\text{-value} = 0.36756$
Conclusion: The mean cost could be $1, or less. At the 5% significance level, there is insufficient evidence to conclude that the mean price of a box of macaroni and cheese is more than $1.
$(0.8291, 1.241)$

"William Shakespeare: The Tragedy of Hamlet, Prince of Denmark," by Jacqueline Ghodsi

THE CHARACTERS (in order of appearance):

HAMLET, Prince of Denmark and student of Statistics
POLONIUS, Hamlet’s tutor
HOROTIO, friend to Hamlet and fellow student

Scene: The great library of the castle, in which Hamlet does his lessons

(The day is fair, but the face of Hamlet is clouded. He paces the large room. His tutor, Polonius, is reprimanding Hamlet regarding the latter’s recent experience. Horatio is seated at the large table at right stage.)

POLONIUS: My Lord, how cans’t thou admit that thou hast seen a ghost! It is but a figment of your imagination!

HAMLET: I beg to differ; I know of a certainty that five-and-seventy in one hundred of us, condemned to the whips and scorns of time as we are, have gazed upon a spirit of health, or goblin damn’d, be their intents wicked or charitable.

POLONIUS If thou doest insist upon thy wretched vision then let me invest your time; be true to thy work and speak to me through the reason of the null and alternate hypotheses. (He turns to Horatio.) Did not Hamlet himself say, “What piece of work is man, how noble in reason, how infinite in faculties? Then let not this foolishness persist. Go, Horatio, make a survey of three-and-sixty and discover what the true proportion be. For my part, I will never succumb to this fantasy, but deem man to be devoid of all reason should thy proposal of at least five-and-seventy in one hundred hold true.

HORATIO (to Hamlet): What should we do, my Lord?

HAMLET: Go to thy purpose, Horatio.

HORATIO: To what end, my Lord?

HAMLET: That you must teach me. But let me conjure you by the rights of our fellowship, by the consonance of our youth, but the obligation of our ever-preserved love, be even and direct with me, whether I am right or no.

(Horatio exits, followed by Polonius, leaving Hamlet to ponder alone.)

(The next day, Hamlet awaits anxiously the presence of his friend, Horatio. Polonius enters and places some books upon the table just a moment before Horatio enters.)

POLONIUS: So, Horatio, what is it thou didst reveal through thy deliberations?

HORATIO: In a random survey, for which purpose thou thyself sent me forth, I did discover that one-and-forty believe fervently that the spirits of the dead walk with us. Before my God, I might not this believe, without the sensible and true avouch of mine own eyes.

POLONIUS: Give thine own thoughts no tongue, Horatio. (Polonius turns to Hamlet.) But look to’t I charge you, my Lord. Come Horatio, let us go together, for this is not our test. (Horatio and Polonius leave together.)

HAMLET: To reject, or not reject, that is the question: whether ‘tis nobler in the mind to suffer the slings and arrows of outrageous statistics, or to take arms against a sea of data, and, by opposing, end them. (Hamlet resignedly attends to his task.)

(Curtain falls)

"Untitled," by Stephen Chen

I've often wondered how software is released and sold to the public. Ironically, I work for a company that sells products with known problems. Unfortunately, most of the problems are difficult to create, which makes them difficult to fix. I usually use the test program X, which tests the product, to try to create a specific problem. When the test program is run to make an error occur, the likelihood of generating an error is 1%.

So, armed with this knowledge, I wrote a new test program Y that will generate the same error that test program X creates, but more often. To find out if my test program is better than the original, so that I can convince the management that I'm right, I ran my test program to find out how often I can generate the same error. When I ran my test program 50 times, I generated the error twice. While this may not seem much better, I think that I can convince the management to use my test program instead of the original test program. Am I right?

$H_{0}: p = 0.01$
$H_{a}: p > 0.01$
Let $P′ =$ the proportion of errors generated
Normal for a single proportion
Decision: Reject the null hypothesis
Conclusion: At the 5% significance level, there is sufficient evidence to conclude that the proportion of errors generated is more than 0.01.

The“plus-4s” confidence interval is $(0.004, 0.144)$.

"Japanese Girls’ Names"

by Kumi Furuichi

It used to be very typical for Japanese girls’ names to end with “ko.” (The trend might have started around my grandmothers’ generation and its peak might have been around my mother’s generation.) “Ko” means “child” in Chinese characters. Parents would name their daughters with “ko” attaching to other Chinese characters which have meanings that they want their daughters to become, such as Sachiko—happy child, Yoshiko—a good child, Yasuko—a healthy child, and so on.

However, I noticed recently that only two out of nine of my Japanese girlfriends at this school have names which end with “ko.” More and more, parents seem to have become creative, modernized, and, sometimes, westernized in naming their children.

I have a feeling that, while 70 percent or more of my mother’s generation would have names with “ko” at the end, the proportion has dropped among my peers. I wrote down all my Japanese friends’, ex-classmates’, co-workers, and acquaintances’ names that I could remember. Following are the names. (Some are repeats.) Test to see if the proportion has dropped for this generation.

Ai, Akemi, Akiko, Ayumi, Chiaki, Chie, Eiko, Eri, Eriko, Fumiko, Harumi, Hitomi, Hiroko, Hiroko, Hidemi, Hisako, Hinako, Izumi, Izumi, Junko, Junko, Kana, Kanako, Kanayo, Kayo, Kayoko, Kazumi, Keiko, Keiko, Kei, Kumi, Kumiko, Kyoko, Kyoko, Madoka, Maho, Mai, Maiko, Maki, Miki, Miki, Mikiko, Mina, Minako, Miyako, Momoko, Nana, Naoko, Naoko, Naoko, Noriko, Rieko, Rika, Rika, Rumiko, Rei, Reiko, Reiko, Sachiko, Sachiko, Sachiyo, Saki, Sayaka, Sayoko, Sayuri, Seiko, Shiho, Shizuka, Sumiko, Takako, Takako, Tomoe, Tomoe, Tomoko, Touko, Yasuko, Yasuko, Yasuyo, Yoko, Yoko, Yoko, Yoshiko, Yoshiko, Yoshiko, Yuka, Yuki, Yuki, Yukiko, Yuko, Yuko.

"Phillip’s Wish," by Suzanne Osorio

My nephew likes to play

Chasing the girls makes his day.

He asked his mother

If it is okay

To get his ear pierced.

She said, “No way!”

To poke a hole through your ear,

Is not what I want for you, dear.

He argued his point quite well,

Says even my macho pal, Mel,

Has gotten this done.

It’s all just for fun.

C’mon please, mom, please, what the hell.

Again Phillip complained to his mother,

Saying half his friends (including their brothers)

Are piercing their ears

And they have no fears

He wants to be like the others.

She said, “I think it’s much less.

We must do a hypothesis test.

And if you are right,

I won’t put up a fight.

But, if not, then my case will rest.”

We proceeded to call fifty guys

To see whose prediction would fly.

Nineteen of the fifty

Said piercing was nifty

And earrings they’d occasionally buy.

Then there’s the other thirty-one,

Who said they’d never have this done.

So now this poem’s finished.

Will his hopes be diminished,

Or will my nephew have his fun?

$H_{0}: p = 0.50$
$H_{a}: p < 0.50$
Let $P′ =$ the proportion of friends that has a pierced ear.
–1.70
$p\text{-value} = 0.0448$
Reason for decision: The $p\text{-value}$ is less than 0.05. (However, they are very close.)
Conclusion: There is sufficient evidence to support the claim that less than 50% of his friends have pierced ears.
Confidence Interval: $(0.245, 0.515)$: The “plus-4s” confidence interval is $(0.259, 0.519)$.

"The Craven," by Mark Salangsang

Once upon a morning dreary

In stats class I was weak and weary.

Pondering over last night’s homework

Whose answers were now on the board

This I did and nothing more.

While I nodded nearly napping

Suddenly, there came a tapping.

As someone gently rapping,

Rapping my head as I snore.

Quoth the teacher, “Sleep no more.”

“In every class you fall asleep,”

The teacher said, his voice was deep.

“So a tally I’ve begun to keep

Of every class you nap and snore.

The percentage being forty-four.”

“My dear teacher I must confess,

While sleeping is what I do best.

The percentage, I think, must be less,

A percentage less than forty-four.”

This I said and nothing more.

“We’ll see,” he said and walked away,

And fifty classes from that day

He counted till the month of May

The classes in which I napped and snored.

The number he found was twenty-four.

At a significance level of 0.05,

Please tell me am I still alive?

Or did my grade just take a dive

Plunging down beneath the floor?

Upon thee I hereby implore.

Toastmasters International cites a report by Gallop Poll that 40% of Americans fear public speaking. A student believes that less than 40% of students at her school fear public speaking. She randomly surveys 361 schoolmates and finds that 135 report they fear public speaking. Conduct a hypothesis test to determine if the percent at her school is less than 40%.

$H_{0}: p = 0.40$
$H_{a}: p < 0.40$
Let $P′ =$ the proportion of schoolmates who fear public speaking.
–1.01
$p\text{-value} = 0.1563$
Conclusion: There is insufficient evidence to support the claim that less than 40% of students at the school fear public speaking.
Confidence Interval: $(0.3241, 0.4240)$: The “plus-4s” confidence interval is $(0.3257, 0.4250)$.

Sixty-eight percent of online courses taught at community colleges nationwide were taught by full-time faculty. To test if 68% also represents California’s percent for full-time faculty teaching the online classes, Long Beach City College (LBCC) in California, was randomly selected for comparison. In the same year, 34 of the 44 online courses LBCC offered were taught by full-time faculty. Conduct a hypothesis test to determine if 68% represents California. NOTE: For more accurate results, use more California community colleges and this past year's data.

According to an article in Bloomberg Businessweek , New York City's most recent adult smoking rate is 14%. Suppose that a survey is conducted to determine this year’s rate. Nine out of 70 randomly chosen N.Y. City residents reply that they smoke. Conduct a hypothesis test to determine if the rate is still 14% or if it has decreased.

$H_{0}: p = 0.14$
$H_{a}: p < 0.14$
Let $P′ =$ the proportion of NYC residents that smoke.
–0.2756
$p\text{-value} = 0.3914$
At the 5% significance level, there is insufficient evidence to conclude that the proportion of NYC residents who smoke is less than 0.14.
Confidence Interval: $(0.0502, 0.2070)$: The “plus-4s” confidence interval (see chapter 8) is $(0.0676, 0.2297)$.

The mean age of De Anza College students in a previous term was 26.6 years old. An instructor thinks the mean age for online students is older than 26.6. She randomly surveys 56 online students and finds that the sample mean is 29.4 with a standard deviation of 2.1. Conduct a hypothesis test.

Registered nurses earned an average annual salary of $69,110. For that same year, a survey was conducted of 41 California registered nurses to determine if the annual salary is higher than $69,110 for California nurses. The sample average was $71,121 with a sample standard deviation of $7,489. Conduct a hypothesis test.

$H_{0}: \mu = 69,110$
$H_{0}: \mu > 69,110$
Let $\bar{X} =$ the mean salary in dollars for California registered nurses.
$t = 1.719$
$p\text{-value}: 0.0466$
Conclusion: At the 5% significance level, there is sufficient evidence to conclude that the mean salary of California registered nurses exceeds $69,110.
$($68,757, $73,485)$

La Leche League International reports that the mean age of weaning a child from breastfeeding is age four to five worldwide. In America, most nursing mothers wean their children much earlier. Suppose a random survey is conducted of 21 U.S. mothers who recently weaned their children. The mean weaning age was nine months (3/4 year) with a standard deviation of 4 months. Conduct a hypothesis test to determine if the mean weaning age in the U.S. is less than four years old.

After conducting the test, your decision and conclusion are

Reject $H_{0}$: There is sufficient evidence to conclude that more than 30% of teen girls smoke to stay thin.
Do not reject $H_{0}$: There is not sufficient evidence to conclude that less than 30% of teen girls smoke to stay thin.
Do not reject $H_{0}$: There is not sufficient evidence to conclude that more than 30% of teen girls smoke to stay thin.
Reject $H_{0}$: There is sufficient evidence to conclude that less than 30% of teen girls smoke to stay thin.

At a 1% level of significance, an appropriate conclusion is:

There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than 20%.
There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is more than 20%.
There is sufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is less than 20%.
There is insufficient evidence to conclude that the percent of EVC students who attended the midnight showing of Harry Potter is at least 20%.

At a significance level of $a = 0.05$, what is the correct conclusion?

There is enough evidence to conclude that the mean number of hours is more than 4.75
There is enough evidence to conclude that the mean number of hours is more than 4.5
There is not enough evidence to conclude that the mean number of hours is more than 4.5
There is not enough evidence to conclude that the mean number of hours is more than 4.75

Instructions: For the following ten exercises,

Hypothesis testing: For the following ten exercises, answer each question.

State the null and alternate hypothesis.

State the $p\text{-value}$.

State $\alpha$.

What is your decision?

Write a conclusion.

Answer any other questions asked in the problem.

According to the Center for Disease Control website, in 2011 at least 18% of high school students have smoked a cigarette. An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized–approximately 1,200 students–small city demographic) to determine if the local high school’s percentage was lower. One hundred fifty students were chosen at random and surveyed. Of the 150 students surveyed, 82 have smoked. Use a significance level of 0.05 and using appropriate statistical evidence, conduct a hypothesis test and state the conclusions.

A recent survey in the N.Y. Times Almanac indicated that 48.8% of families own stock. A broker wanted to determine if this survey could be valid. He surveyed a random sample of 250 families and found that 142 owned some type of stock. At the 0.05 significance level, can the survey be considered to be accurate?

$H_{0}: p = 0.488$ $H_{a}: p \neq 0.488$
$p\text{-value} = 0.0114$
$\alpha = 0.05$
Reject the null hypothesis.
At the 5% level of significance, there is enough evidence to conclude that 48.8% of families own stocks.
The survey does not appear to be accurate.

Driver error can be listed as the cause of approximately 54% of all fatal auto accidents, according to the American Automobile Association. Thirty randomly selected fatal accidents are examined, and it is determined that 14 were caused by driver error. Using $\alpha = 0.05$, is the AAA proportion accurate?

The US Department of Energy reported that 51.7% of homes were heated by natural gas. A random sample of 221 homes in Kentucky found that 115 were heated by natural gas. Does the evidence support the claim for Kentucky at the $\alpha = 0.05$ level in Kentucky? Are the results applicable across the country? Why?

$H_{0}: p = 0.517$ $H_{0}: p \neq 0.517$
$p\text{-value} = 0.9203$.
$\alpha = 0.05$.
Do not reject the null hypothesis.
At the 5% significance level, there is not enough evidence to conclude that the proportion of homes in Kentucky that are heated by natural gas is 0.517.
However, we cannot generalize this result to the entire nation. First, the sample’s population is only the state of Kentucky. Second, it is reasonable to assume that homes in the extreme north and south will have extreme high usage and low usage, respectively. We would need to expand our sample base to include these possibilities if we wanted to generalize this claim to the entire nation.

For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. The library director in Owensboro, Kentucky feels this is not true, so she asked a local college statistic class to conduct a survey. The class randomly selected 100 patrons and found that 82 borrowed books. Did the class demonstrate that the percentage was higher in Owensboro, KY? Use $\alpha = 0.01$ level of significance. What is the possible proportion of patrons that do borrow books from the Owensboro Library?

The Weather Underground reported that the mean amount of summer rainfall for the northeastern US is at least 11.52 inches. Ten cities in the northeast are randomly selected and the mean rainfall amount is calculated to be 7.42 inches with a standard deviation of 1.3 inches. At the $\alpha = 0.05 level$, can it be concluded that the mean rainfall was below the reported average? What if $\alpha = 0.01$? Assume the amount of summer rainfall follows a normal distribution.

$H_{0}: \mu \geq 11.52$ $H_{a}: \mu < 11.52$
$p\text{-value} = 0.000002$ which is almost 0.
At the 5% significance level, there is enough evidence to conclude that the mean amount of summer rain in the northeaster US is less than 11.52 inches, on average.
We would make the same conclusion if alpha was 1% because the $p\text{-value}$ is almost 0.

A survey in the N.Y. Times Almanac finds the mean commute time (one way) is 25.4 minutes for the 15 largest US cities. The Austin, TX chamber of commerce feels that Austin’s commute time is less and wants to publicize this fact. The mean for 25 randomly selected commuters is 22.1 minutes with a standard deviation of 5.3 minutes. At the $\alpha = 0.10$ level, is the Austin, TX commute significantly less than the mean commute time for the 15 largest US cities?

A report by the Gallup Poll found that a woman visits her doctor, on average, at most 5.8 times each year. A random sample of 20 women results in these yearly visit totals

3; 2; 1; 3; 7; 2; 9; 4; 6; 6; 8; 0; 5; 6; 4; 2; 1; 3; 4; 1

At the $\alpha = 0.05$ level can it be concluded that the sample mean is higher than 5.8 visits per year?

$H_{0}: \mu \leq 5.8$ $H_{a}: \mu > 5.8$
$p\text{-value} = 0.9987$
At the 5% level of significance, there is not enough evidence to conclude that a woman visits her doctor, on average, more than 5.8 times a year.

According to the N.Y. Times Almanac the mean family size in the U.S. is 3.18. A sample of a college math class resulted in the following family sizes:

5; 4; 5; 4; 4; 3; 6; 4; 3; 3; 5; 5; 6; 3; 3; 2; 7; 4; 5; 2; 2; 2; 3; 2

At $\alpha = 0.05$ level, is the class’ mean family size greater than the national average? Does the Almanac result remain valid? Why?

The student academic group on a college campus claims that freshman students study at least 2.5 hours per day, on average. One Introduction to Statistics class was skeptical. The class took a random sample of 30 freshman students and found a mean study time of 137 minutes with a standard deviation of 45 minutes. At α = 0.01 level, is the student academic group’s claim correct?

$H_{0}: \mu \geq 150$ $H_{0}: \mu < 150$
$p\text{-value} = 0.0622$
$\alpha = 0.01$
At the 1% significance level, there is not enough evidence to conclude that freshmen students study less than 2.5 hours per day, on average.
The student academic group’s claim appears to be correct.

9.7: Hypothesis Testing of a Single Mean and Single Proportion

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Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, a complete guide to chi-square test, a complete guide on hypothesis testing in statistics, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, a comprehensive guide to understand mean squared error, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution.

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Lesson 10 of 24 By Avijeet Biswal

A Complete Guide on Hypothesis Testing in Statistics

In today’s data-driven world , decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life -

A teacher assumes that 60% of his college's students come from lower-middle-class families.
A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

Here, x̅ is the sample mean,
μ0 is the population mean,
σ is the standard deviation,
n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average.

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Step 1: specify your null and alternate hypotheses.

It is critical to rephrase your original research hypothesis (the prediction that you wish to study) as a null (Ho) and alternative (Ha) hypothesis so that you can test it quantitatively. Your first hypothesis, which predicts a link between variables, is generally your alternate hypothesis. The null hypothesis predicts no link between the variables of interest.

Step 2: Gather Data

For a statistical test to be legitimate, sampling and data collection must be done in a way that is meant to test your hypothesis. You cannot draw statistical conclusions about the population you are interested in if your data is not representative.

Step 3: Conduct a Statistical Test

Other statistical tests are available, but they all compare within-group variance (how to spread out the data inside a category) against between-group variance (how different the categories are from one another). If the between-group variation is big enough that there is little or no overlap between groups, your statistical test will display a low p-value to represent this. This suggests that the disparities between these groups are unlikely to have occurred by accident. Alternatively, if there is a large within-group variance and a low between-group variance, your statistical test will show a high p-value. Any difference you find across groups is most likely attributable to chance. The variety of variables and the level of measurement of your obtained data will influence your statistical test selection.

Step 4: Determine Rejection Of Your Null Hypothesis

Your statistical test results must determine whether your null hypothesis should be rejected or not. In most circumstances, you will base your judgment on the p-value provided by the statistical test. In most circumstances, your preset level of significance for rejecting the null hypothesis will be 0.05 - that is, when there is less than a 5% likelihood that these data would be seen if the null hypothesis were true. In other circumstances, researchers use a lower level of significance, such as 0.01 (1%). This reduces the possibility of wrongly rejecting the null hypothesis.

Step 5: Present Your Results

The findings of hypothesis testing will be discussed in the results and discussion portions of your research paper, dissertation, or thesis. You should include a concise overview of the data and a summary of the findings of your statistical test in the results section. You can talk about whether your results confirmed your initial hypothesis or not in the conversation. Rejecting or failing to reject the null hypothesis is a formal term used in hypothesis testing. This is likely a must for your statistics assignments.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

The null hypothesis is (H0 <= 90) or less change.
A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true].

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

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A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales. If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

Why is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore Simplilearn’s Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is hypothesis testing and its types?

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating two hypotheses: the null hypothesis (H0), which represents the default assumption, and the alternative hypothesis (Ha), which contradicts H0. The goal is to assess the evidence and determine whether there is enough statistical significance to reject the null hypothesis in favor of the alternative hypothesis.

Types of hypothesis testing:

One-sample test: Used to compare a sample to a known value or a hypothesized value.
Two-sample test: Compares two independent samples to assess if there is a significant difference between their means or distributions.
Paired-sample test: Compares two related samples, such as pre-test and post-test data, to evaluate changes within the same subjects over time or under different conditions.
Chi-square test: Used to analyze categorical data and determine if there is a significant association between variables.
ANOVA (Analysis of Variance): Compares means across multiple groups to check if there is a significant difference between them.

3. What are the steps of hypothesis testing?

The steps of hypothesis testing are as follows:

Formulate the hypotheses: State the null hypothesis (H0) and the alternative hypothesis (Ha) based on the research question.
Set the significance level: Determine the acceptable level of error (alpha) for making a decision.
Collect and analyze data: Gather and process the sample data.
Compute test statistic: Calculate the appropriate statistical test to assess the evidence.
Make a decision: Compare the test statistic with critical values or p-values and determine whether to reject H0 in favor of Ha or not.
Draw conclusions: Interpret the results and communicate the findings in the context of the research question.

4. What are the 2 types of hypothesis testing?

One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the author.

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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Fundamental Analysis

Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example

What Is Hypothesis Testing?

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population, or from a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
The test provides evidence concerning the plausibility of the hypothesis, given the data.
Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an analyst tests a statistical sample, with the goal of providing evidence on the plausibility of the null hypothesis.

Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis (e.g., the population mean return is not equal to zero). Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

4 Steps of Hypothesis Testing

All hypotheses are tested using a four-step process:

The first step is for the analyst to state the hypotheses.
The second step is to formulate an analysis plan, which outlines how the data will be evaluated.
The third step is to carry out the plan and analyze the sample data.
The final step is to analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Real-World Example of Hypothesis Testing

If, for example, a person wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct.

Mathematically, the null hypothesis would be represented as Ho: P = 0.5. The alternative hypothesis would be denoted as "Ha" and be identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is then tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If, on the other hand, there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

Some staticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What is Hypothesis Testing?

Hypothesis testing refers to a process used by analysts to assess the plausibility of a hypothesis by using sample data. In hypothesis testing, statisticians formulate two hypotheses: the null hypothesis and the alternative hypothesis. A null hypothesis determines there is no difference between two groups or conditions, while the alternative hypothesis determines that there is a difference. Researchers evaluate the statistical significance of the test based on the probability that the null hypothesis is true.

What are the Four Key Steps Involved in Hypothesis Testing?

Hypothesis testing begins with an analyst stating two hypotheses, with only one that can be right. The analyst then formulates an analysis plan, which outlines how the data will be evaluated. Next, they move to the testing phase and analyze the sample data. Finally, the analyst analyzes the results and either rejects the null hypothesis or states that the null hypothesis is plausible, given the data.

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

The Bottom Line

Hypothesis testing refers to a statistical process that helps researchers and/or analysts determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. There are different types of hypothesis testing, each with their own set of rules and procedures. However, all hypothesis testing methods have the same four step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result. Hypothesis testing plays a vital part of the scientific process, helping to test assumptions and make better data-based decisions.

Sage. " Introduction to Hypothesis Testing. " Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples. "

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Statistics and probability

Course: statistics and probability > unit 12, hypothesis testing and p-values.

One-tailed and two-tailed tests
Z-statistics vs. T-statistics
Small sample hypothesis test
Large sample proportion hypothesis testing

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Knowledge Base

An Introduction to t Tests | Definitions, Formula and Examples

Published on January 31, 2020 by Rebecca Bevans . Revised on June 22, 2023.

A t test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another.

The null hypothesis ( H 0 ) is that the true difference between these group means is zero.
The alternate hypothesis ( H a ) is that the true difference is different from zero.

When to use a t test, what type of t test should i use, performing a t test, interpreting test results, presenting the results of a t test, other interesting articles, frequently asked questions about t tests.

A t test can only be used when comparing the means of two groups (a.k.a. pairwise comparison). If you want to compare more than two groups, or if you want to do multiple pairwise comparisons, use an ANOVA test or a post-hoc test.

The t test is a parametric test of difference, meaning that it makes the same assumptions about your data as other parametric tests. The t test assumes your data:

are independent
are (approximately) normally distributed
have a similar amount of variance within each group being compared (a.k.a. homogeneity of variance)

If your data do not fit these assumptions, you can try a nonparametric alternative to the t test, such as the Wilcoxon Signed-Rank test for data with unequal variances .

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When choosing a t test, you will need to consider two things: whether the groups being compared come from a single population or two different populations, and whether you want to test the difference in a specific direction.

One-sample, two-sample, or paired t test?

If the groups come from a single population (e.g., measuring before and after an experimental treatment), perform a paired t test . This is a within-subjects design .
If the groups come from two different populations (e.g., two different species, or people from two separate cities), perform a two-sample t test (a.k.a. independent t test ). This is a between-subjects design .
If there is one group being compared against a standard value (e.g., comparing the acidity of a liquid to a neutral pH of 7), perform a one-sample t test .

One-tailed or two-tailed t test?

If you only care whether the two populations are different from one another, perform a two-tailed t test .
If you want to know whether one population mean is greater than or less than the other, perform a one-tailed t test.
Your observations come from two separate populations (separate species), so you perform a two-sample t test.
You don’t care about the direction of the difference, only whether there is a difference, so you choose to use a two-tailed t test.

The t test estimates the true difference between two group means using the ratio of the difference in group means over the pooled standard error of both groups. You can calculate it manually using a formula, or use statistical analysis software.

T test formula

The formula for the two-sample t test (a.k.a. the Student’s t-test) is shown below.

$\begin{equation*}t=\dfrac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{(s^2(\frac{1}{n_{1}}+\frac{1}{n_{2}}))}}}\end{equation*}$

In this formula, t is the t value, x 1 and x 2 are the means of the two groups being compared, s 2 is the pooled standard error of the two groups, and n 1 and n 2 are the number of observations in each of the groups.

A larger t value shows that the difference between group means is greater than the pooled standard error, indicating a more significant difference between the groups.

You can compare your calculated t value against the values in a critical value chart (e.g., Student’s t table) to determine whether your t value is greater than what would be expected by chance. If so, you can reject the null hypothesis and conclude that the two groups are in fact different.

T test function in statistical software

Most statistical software (R, SPSS, etc.) includes a t test function. This built-in function will take your raw data and calculate the t value. It will then compare it to the critical value, and calculate a p -value . This way you can quickly see whether your groups are statistically different.

In your comparison of flower petal lengths, you decide to perform your t test using R. The code looks like this:

Download the data set to practice by yourself.

Sample data set

If you perform the t test for your flower hypothesis in R, you will receive the following output:

The output provides:

An explanation of what is being compared, called data in the output table.
The t value : -33.719. Note that it’s negative; this is fine! In most cases, we only care about the absolute value of the difference, or the distance from 0. It doesn’t matter which direction.
The degrees of freedom : 30.196. Degrees of freedom is related to your sample size, and shows how many ‘free’ data points are available in your test for making comparisons. The greater the degrees of freedom, the better your statistical test will work.
The p value : 2.2e-16 (i.e. 2.2 with 15 zeros in front). This describes the probability that you would see a t value as large as this one by chance.
A statement of the alternative hypothesis ( H a ). In this test, the H a is that the difference is not 0.
The 95% confidence interval . This is the range of numbers within which the true difference in means will be 95% of the time. This can be changed from 95% if you want a larger or smaller interval, but 95% is very commonly used.
The mean petal length for each group.

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When reporting your t test results, the most important values to include are the t value , the p value , and the degrees of freedom for the test. These will communicate to your audience whether the difference between the two groups is statistically significant (a.k.a. that it is unlikely to have happened by chance).

You can also include the summary statistics for the groups being compared, namely the mean and standard deviation . In R, the code for calculating the mean and the standard deviation from the data looks like this:

flower.data %>% group_by(Species) %>% summarize(mean_length = mean(Petal.Length), sd_length = sd(Petal.Length))

In our example, you would report the results like this:

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

Chi square test of independence
Statistical power
Descriptive statistics
Degrees of freedom
Pearson correlation
Null hypothesis

Methodology

Double-blind study
Case-control study
Research ethics
Data collection
Hypothesis testing
Structured interviews

Research bias

Hawthorne effect
Unconscious bias
Recall bias
Halo effect
Self-serving bias
Information bias

A t-test is a statistical test that compares the means of two samples . It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.

A t-test measures the difference in group means divided by the pooled standard error of the two group means.

In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance (p-value).

Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test .

If you want to know only whether a difference exists, use a two-tailed test . If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test .

A one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).

A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time (for example, measuring student performance on a test before and after being taught the material).

A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.

If you want to compare the means of several groups at once, it’s best to use another statistical test such as ANOVA or a post-hoc test.

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9.4 Full Hypothesis Test Examples

Tests on means, example 9.8.

Jeffrey, as an eight-year old, established a mean time of 16.43 seconds for swimming the 25-yard freestyle, with a standard deviation of 0.8 seconds . His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster using goggles. Frank bought Jeffrey a new pair of expensive goggles and timed Jeffrey for 15 25-yard freestyle swims . For the 15 swims, Jeffrey's mean time was 16 seconds. Frank thought that the goggles helped Jeffrey to swim faster than the 16.43 seconds. Conduct a hypothesis test using a preset α = 0.05.

Set up the Hypothesis Test:

Since the problem is about a mean, this is a test of a single population mean .

Set the null and alternative hypothesis:

In this case there is an implied challenge or claim. This is that the goggles will reduce the swimming time. The effect of this is to set the hypothesis as a one-tailed test. The claim will always be in the alternative hypothesis because the burden of proof always lies with the alternative. Remember that the status quo must be defeated with a high degree of confidence, in this case 95 % confidence. The null and alternative hypotheses are thus:

H 0 : μ ≥ 16.43 H a : μ < 16.43

For Jeffrey to swim faster, his time will be less than 16.43 seconds. The "<" tells you this is left-tailed.

Determine the distribution needed:

Random variable: X ¯ X ¯ = the mean time to swim the 25-yard freestyle.

Distribution for the test statistic:

The sample size is less than 30 and we do not know the population standard deviation so this is a t-test. and the proper formula is: t c = X ¯ - μ 0 σ / n t c = X ¯ - μ 0 σ / n

μ 0 = 16.43 comes from H 0 and not the data. X ¯ X ¯ = 16. s = 0.8, and n = 15.

Our step 2, setting the level of significance, has already been determined by the problem, .05 for a 95 % significance level. It is worth thinking about the meaning of this choice. The Type I error is to conclude that Jeffrey swims the 25-yard freestyle, on average, in less than 16.43 seconds when, in fact, he actually swims the 25-yard freestyle, on average, in 16.43 seconds. (Reject the null hypothesis when the null hypothesis is true.) For this case the only concern with a Type I error would seem to be that Jeffery’s dad may fail to bet on his son’s victory because he does not have appropriate confidence in the effect of the goggles.

To find the critical value we need to select the appropriate test statistic. We have concluded that this is a t-test on the basis of the sample size and that we are interested in a population mean. We can now draw the graph of the t-distribution and mark the critical value. For this problem the degrees of freedom are n-1, or 14. Looking up 14 degrees of freedom at the 0.05 column of the t-table we find 1.761. This is the critical value and we can put this on our graph.

Step 3 is the calculation of the test statistic using the formula we have selected. We find that the calculated test statistic is 2.08, meaning that the sample mean is 2.08 standard deviations away from the hypothesized mean of 16.43.

Step 4 has us compare the test statistic and the critical value and mark these on the graph. We see that the test statistic is in the tail and thus we move to step 4 and reach a conclusion. The probability that an average time of 16 minutes could come from a distribution with a population mean of 16.43 minutes is too unlikely for us to accept the null hypothesis. We cannot accept the null.

Step 5 has us state our conclusions first formally and then less formally. A formal conclusion would be stated as: “With a 95% level of significance we cannot accept the null hypothesis that the swimming time with goggles comes from a distribution with a population mean time of 16.43 minutes.” Less formally, “With 95% significance we believe that the goggles improves swimming speed”

If we wished to use the p-value system of reaching a conclusion we would calculate the statistic and take the additional step to find the probability of being 2.08 standard deviations from the mean on a t-distribution. This value is .0187. Comparing this to the α-level of .05 we see that we cannot accept the null. The p-value has been put on the graph as the shaded area beyond -2.08 and it shows that it is smaller than the hatched area which is the alpha level of 0.05. Both methods reach the same conclusion that we cannot accept the null hypothesis.

The mean throwing distance of a football for Marco, a high school freshman quarterback, is 40 yards, with a standard deviation of two yards. The team coach tells Marco to adjust his grip to get more distance. The coach records the distances for 20 throws. For the 20 throws, Marco’s mean distance was 45 yards. The coach thought the different grip helped Marco throw farther than 40 yards. Conduct a hypothesis test using a preset α = 0.05. Assume the throw distances for footballs are normal.

First, determine what type of test this is, set up the hypothesis test, find the p -value, sketch the graph, and state your conclusion.

Example 9.9

Jane has just begun her new job as on the sales force of a very competitive company. In a sample of 16 sales calls it was found that she closed the contract for an average value of 108 dollars with a standard deviation of 12 dollars. Test at 5% significance that the population mean is at least 100 dollars against the alternative that it is less than 100 dollars. Company policy requires that new members of the sales force must exceed an average of $100 per contract during the trial employment period. Can we conclude that Jane has met this requirement at the significance level of 95%?

H 0 : µ ≤ 100 H a : µ > 100 The null and alternative hypothesis are for the parameter µ because the number of dollars of the contracts is a continuous random variable. Also, this is a one-tailed test because the company has only an interested if the number of dollars per contact is below a particular number not "too high" a number. This can be thought of as making a claim that the requirement is being met and thus the claim is in the alternative hypothesis.
Test statistic: t c = x ¯ − µ 0 s n = 108 − 100 ( 12 16 ) = 2.67 t c = x ¯ − µ 0 s n = 108 − 100 ( 12 16 ) = 2.67
Critical value: t a = 1.753 t a = 1.753 with n-1 degrees of freedom= 15

The test statistic is a Student's t because the sample size is below 30; therefore, we cannot use the normal distribution. Comparing the calculated value of the test statistic and the critical value of t t ( t a ) ( t a ) at a 5% significance level, we see that the calculated value is in the tail of the distribution. Thus, we conclude that 108 dollars per contract is significantly larger than the hypothesized value of 100 and thus we cannot accept the null hypothesis. There is evidence that supports Jane's performance meets company standards.

It is believed that a stock price for a particular company will grow at a rate of $5 per week with a standard deviation of $1. An investor believes the stock won’t grow as quickly. The changes in stock price is recorded for ten weeks and are as follows: $4, $3, $2, $3, $1, $7, $2, $1, $1, $2. Perform a hypothesis test using a 5% level of significance. State the null and alternative hypotheses, state your conclusion, and identify the Type I errors.

Example 9.10

A manufacturer of salad dressings uses machines to dispense liquid ingredients into bottles that move along a filling line. The machine that dispenses salad dressings is working properly when 8 ounces are dispensed. Suppose that the average amount dispensed in a particular sample of 35 bottles is 7.91 ounces with a variance of 0.03 ounces squared, s 2 s 2 . Is there evidence that the machine should be stopped and production wait for repairs? The lost production from a shutdown is potentially so great that management feels that the level of significance in the analysis should be 99%.

Again we will follow the steps in our analysis of this problem.

STEP 1 : Set the Null and Alternative Hypothesis. The random variable is the quantity of fluid placed in the bottles. This is a continuous random variable and the parameter we are interested in is the mean. Our hypothesis therefore is about the mean. In this case we are concerned that the machine is not filling properly. From what we are told it does not matter if the machine is over-filling or under-filling, both seem to be an equally bad error. This tells us that this is a two-tailed test: if the machine is malfunctioning it will be shutdown regardless if it is from over-filling or under-filling. The null and alternative hypotheses are thus:

STEP 2 : Decide the level of significance and draw the graph showing the critical value.

This problem has already set the level of significance at 99%. The decision seems an appropriate one and shows the thought process when setting the significance level. Management wants to be very certain, as certain as probability will allow, that they are not shutting down a machine that is not in need of repair. To draw the distribution and the critical value, we need to know which distribution to use. Because this is a continuous random variable and we are interested in the mean, and the sample size is greater than 30, the appropriate distribution is the normal distribution and the relevant critical value is 2.575 from the normal table or the t-table at 0.005 column and infinite degrees of freedom. We draw the graph and mark these points.

STEP 3 : Calculate sample parameters and the test statistic. The sample parameters are provided, the sample mean is 7.91 and the sample variance is .03 and the sample size is 35. We need to note that the sample variance was provided not the sample standard deviation, which is what we need for the formula. Remembering that the standard deviation is simply the square root of the variance, we therefore know the sample standard deviation, s, is 0.173. With this information we calculate the test statistic as -3.07, and mark it on the graph.

STEP 4 : Compare test statistic and the critical values Now we compare the test statistic and the critical value by placing the test statistic on the graph. We see that the test statistic is in the tail, decidedly greater than the critical value of 2.575. We note that even the very small difference between the hypothesized value and the sample value is still a large number of standard deviations. The sample mean is only 0.08 ounces different from the required level of 8 ounces, but it is 3 plus standard deviations away and thus we cannot accept the null hypothesis.

STEP 5 : Reach a Conclusion

Three standard deviations of a test statistic will guarantee that the test will fail. The probability that anything is within three standard deviations is almost zero. Actually it is 0.0026 on the normal distribution, which is certainly almost zero in a practical sense. Our formal conclusion would be “ At a 99% level of significance we cannot accept the hypothesis that the sample mean came from a distribution with a mean of 8 ounces” Or less formally, and getting to the point, “At a 99% level of significance we conclude that the machine is under filling the bottles and is in need of repair”.

Hypothesis Test for Proportions

Just as there were confidence intervals for proportions, or more formally, the population parameter p of the binomial distribution, there is the ability to test hypotheses concerning p .

The population parameter for the binomial is p . The estimated value (point estimate) for p is p′ where p′ = x/n , x is the number of successes in the sample and n is the sample size.

When you perform a hypothesis test of a population proportion p , you take a simple random sample from the population. The conditions for a binomial distribution must be met, which are: there are a certain number n of independent trials meaning random sampling, the outcomes of any trial are binary, success or failure, and each trial has the same probability of a success p . The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np′ and nq′ must both be greater than five ( np′ > 5 and nq′ > 5). In this case the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with μ = np μ = np and σ = npq σ = npq . Remember that q = 1 – p q = 1 – p . There is no distribution that can correct for this small sample bias and thus if these conditions are not met we simply cannot test the hypothesis with the data available at that time. We met this condition when we first were estimating confidence intervals for p .

Again, we begin with the standardizing formula modified because this is the distribution of a binomial.

Substituting p 0 p 0 , the hypothesized value of p , we have:

This is the test statistic for testing hypothesized values of p , where the null and alternative hypotheses take one of the following forms:

The decision rule stated above applies here also: if the calculated value of Z c shows that the sample proportion is "too many" standard deviations from the hypothesized proportion, the null hypothesis cannot be accepted. The decision as to what is "too many" is pre-determined by the analyst depending on the level of significance required in the test.

Example 9.11

The mortgage department of a large bank is interested in the nature of loans of first-time borrowers. This information will be used to tailor their marketing strategy. They believe that 50% of first-time borrowers take out smaller loans than other borrowers. They perform a hypothesis test to determine if the percentage is the same or different from 50% . They sample 100 first-time borrowers and find 53 of these loans are smaller that the other borrowers. For the hypothesis test, they choose a 5% level of significance.

STEP 1 : Set the null and alternative hypothesis.

H 0 : p = 0.50 H a : p ≠ 0.50

The words "is the same or different from" tell you this is a two-tailed test. The Type I and Type II errors are as follows: The Type I error is to conclude that the proportion of borrowers is different from 50% when, in fact, the proportion is actually 50%. (Reject the null hypothesis when the null hypothesis is true). The Type II error is there is not enough evidence to conclude that the proportion of first time borrowers differs from 50% when, in fact, the proportion does differ from 50%. (You fail to reject the null hypothesis when the null hypothesis is false.)

STEP 2 : Decide the level of significance and draw the graph showing the critical value

The level of significance has been set by the problem at the 95% level. Because this is two-tailed test one-half of the alpha value will be in the upper tail and one-half in the lower tail as shown on the graph. The critical value for the normal distribution at the 95% level of confidence is 1.96. This can easily be found on the student’s t-table at the very bottom at infinite degrees of freedom remembering that at infinity the t-distribution is the normal distribution. Of course the value can also be found on the normal table but you have go looking for one-half of 95 (0.475) inside the body of the table and then read out to the sides and top for the number of standard deviations.

STEP 3 : Calculate the sample parameters and critical value of the test statistic.

The test statistic is a normal distribution, Z, for testing proportions and is:

For this case, the sample of 100 found 53 first-time borrowers were different from other borrowers. The sample proportion, p′ = 53/100= 0.53 The test question, therefore, is : “Is 0.53 significantly different from .50?” Putting these values into the formula for the test statistic we find that 0.53 is only 0.60 standard deviations away from .50. This is barely off of the mean of the standard normal distribution of zero. There is virtually no difference from the sample proportion and the hypothesized proportion in terms of standard deviations.

STEP 4 : Compare the test statistic and the critical value.

The calculated value is well within the critical values of ± 1.96 standard deviations and thus we cannot reject the null hypothesis. To reject the null hypothesis we need significant evident of difference between the hypothesized value and the sample value. In this case the sample value is very nearly the same as the hypothesized value measured in terms of standard deviations.

STEP 5 : Reach a conclusion

The formal conclusion would be “At a 95% level of significance we cannot reject the null hypothesis that 50% of first-time borrowers have the same size loans as other borrowers”. Less formally we would say that “There is no evidence that one-half of first-time borrowers are significantly different in loan size from other borrowers”. Notice the length to which the conclusion goes to include all of the conditions that are attached to the conclusion. Statisticians for all the criticism they receive, are careful to be very specific even when this seems trivial. Statisticians cannot say more than they know and the data constrain the conclusion to be within the metes and bounds of the data.

Try It 9.11

A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. She performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.

Example 9.12

Suppose a consumer group suspects that the proportion of households that have three or more cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey 150 households with the result that 43 of the households have three or more cell phones.

Here is an abbreviate version of the system to solve hypothesis tests applied to a test on a proportions.

Example 9.13

The National Institute of Standards and Technology provides exact data on conductivity properties of materials. Following are conductivity measurements for 11 randomly selected pieces of a particular type of glass.

1.11; 1.07; 1.11; 1.07; 1.12; 1.08; .98; .98 1.02; .95; .95 Is there convincing evidence that the average conductivity of this type of glass is greater than one? Use a significance level of 0.05.

Let’s follow a four-step process to answer this statistical question.

H 0 : μ ≤ 1
H a : μ > 1
Plan : We are testing a sample mean without a known population standard deviation with less than 30 observations. Therefore, we need to use a Student's-t distribution. Assume the underlying population is normal.
Do the calculations and draw the graph .
State the Conclusions : We cannot accept the null hypothesis. It is reasonable to state that the data supports the claim that the average conductivity level is greater than one.

Example 9.14

In a study of 420,019 cell phone users, 172 of the subjects developed brain cancer. Test the claim that cell phone users developed brain cancer at a greater rate than that for non-cell phone users (the rate of brain cancer for non-cell phone users is 0.0340%). Since this is a critical issue, use a 0.005 significance level. Explain why the significance level should be so low in terms of a Type I error.

H 0 : p ≤ 0.00034
H a : p > 0.00034

If we commit a Type I error, we are essentially accepting a false claim. Since the claim describes cancer-causing environments, we want to minimize the chances of incorrectly identifying causes of cancer.

We will be testing a sample proportion with x = 172 and n = 420,019. The sample is sufficiently large because we have np' = 420,019(0.00034) = 142.8, nq' = 420,019(0.99966) = 419,876.2, two independent outcomes, and a fixed probability of success p' = 0.00034. Thus we will be able to generalize our results to the population.

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How to Write a Great Hypothesis

Hypothesis Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk, "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis, operational definitions, types of hypotheses, hypotheses examples.

Collecting Data

Frequently Asked Questions

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study.

One hypothesis example would be a study designed to look at the relationship between sleep deprivation and test performance might have a hypothesis that states: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

Forming a question
Performing background research
Creating a hypothesis
Designing an experiment
Collecting data
Analyzing the results
Drawing conclusions
Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. It is only at this point that researchers begin to develop a testable hypothesis. Unless you are creating an exploratory study, your hypothesis should always explain what you expect to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore a number of factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment do not support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk wisdom that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

Is your hypothesis based on your research on a topic?
Can your hypothesis be tested?
Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the journal articles you read . Many authors will suggest questions that still need to be explored.

To form a hypothesis, you should take these steps:

Collect as many observations about a topic or problem as you can.
Evaluate these observations and look for possible causes of the problem.
Create a list of possible explanations that you might want to explore.
After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method , falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that if something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in a number of different ways. One of the basic principles of any type of scientific research is that the results must be replicable. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. How would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

In order to measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming other people. In this situation, the researcher might utilize a simulated task to measure aggressiveness.

Hypothesis Checklist

Does your hypothesis focus on something that you can actually test?
Does your hypothesis include both an independent and dependent variable?
Can you manipulate the variables?
Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

Simple hypothesis : This type of hypothesis suggests that there is a relationship between one independent variable and one dependent variable.
Complex hypothesis : This type of hypothesis suggests a relationship between three or more variables, such as two independent variables and a dependent variable.
Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative sample of the population and then generalizes the findings to the larger group.
Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the dependent variable if you change the independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

"Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
Complex hypothesis: "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."
"Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."

Examples of a complex hypothesis include:

"People with high-sugar diets and sedentary activity levels are more likely to develop depression."
"Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

"Children who receive a new reading intervention will have scores different than students who do not receive the intervention."
"There will be no difference in scores on a memory recall task between children and adults."

Examples of an alternative hypothesis:

"Children who receive a new reading intervention will perform better than students who did not receive the intervention."
"Adults will perform better on a memory task than children."

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as case studies , naturalistic observations , and surveys are often used when it would be impossible or difficult to conduct an experiment . These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a correlational study can then be used to look at how the variables are related. This type of research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually cause another to change.

A Word From Verywell

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Some examples of how to write a hypothesis include:

"Staying up late will lead to worse test performance the next day."
"People who consume one apple each day will visit the doctor fewer times each year."
"Breaking study sessions up into three 20-minute sessions will lead to better test results than a single 60-minute study session."

The four parts of a hypothesis are:

The research question
The independent variable (IV)
The dependent variable (DV)
The proposed relationship between the IV and DV

Castillo M. The scientific method: a need for something better? . AJNR Am J Neuroradiol. 2013;34(9):1669-71. doi:10.3174/ajnr.A3401

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Learn Statistics for Data Science, Machine Learning, and AI – Full Handbook

Karl Pearson was a British mathematician who once said "Statistics is the grammar of science". This holds true especially for Computer and Information Sciences, Physical Science, and Biological Science.

When you are getting started with your journey in Data Science, Data Analytics, Machine Learning, or AI (including Generative AI) having statistical knowledge will help you better leverage data insights and actually understand all the algorithms beyond their implementation approach.

I can't overstate the importance of statistics in data science and Artificial Intelligence. Statistics provides tools and methods to find structure and give deeper data insights. Both Statistics and Mathematics love facts and hate guesses. Knowing the fundamentals of these two important subjects will allow you to think critically, and be creative when using the data to solve business problems and make data-driven decisions.

Key statistical concepts for your data science or data analysis journey with Python Code

In this handbook, I will cover the following Statistics topics for data science, machine learning, and artificial intelligence (including GenAI):

Random variables

Mean, Variance, Standard Deviation

Covariance and Correlation
Probability distribution functions (PDFs)
Bayes Theorem
Linear Regression and Ordinary Least Squares (OLS)

Gauss-Markov Theorem

Parameter properties (Bias, Consistency, Efficiency)
Confidence intervals
Hypothesis testing
Statistical significance
Type I & Type II Error
Statistical tests (Student's t-test, F-test, 2-Sample T-Test, 2-Sample Z-Test, Chi-Square Test)
p-value and its limitations

Inferential Statistics

Central Limit Theorem & Law of Large Numbers
Dimensionality reduction techniques (PCA, FA)
Interview Prep - Top 7 Statistics Questions with Answers
About The Author

How Can You Dive Deeper?

If you have no prior Statistical knowledge and you want to identify and learn the essential statistical concepts from the scratch and prepare for your job interviews, then this handbook is for you. It will also be a good read for anyone who wants to refresh their statistical knowledge.

Prerequisites

Before you start reading this handbook about key concepts in Statistics for Data Science, Machine Learning, and Artificial Intelligence, there are a few prerequisites that will help you make the most out of it.

This list is designed to ensure you are well-prepared and can fully grasp the statistical concepts discussed:

Basic Mathematical Skills : Comfort with high school level mathematics, including algebra and basic calculus, is essential. These skills are crucial for understanding statistical formulas and methods.
Logical Thinking : Ability to think logically and methodically to solve problems will aid in understanding statistical reasoning and applying these concepts to data-driven scenarios.
Computer Literacy : Basic knowledge of using computers and the internet is necessary since many examples and exercises might require the use of statistical software or coding.
Basic knowledge of Python, such as the creation of variables and working with some basic data structures and coding is also required (if you are not familiar with these concepts, check out my Python for Data Science 2024 -Full Course for Beginners here).
Curiosity and Willingness to Learn : A keen interest in learning and exploring data is perhaps the most important prerequisite. The field of data science is constantly evolving, and a proactive approach to learning will be incredibly beneficial.

This handbook assumes no prior knowledge of statistics, making it accessible to beginners. Still, familiarity with the above concepts will greatly enhance your understanding and ability to apply statistical methods effectively in various domains.

If you want to learn Mathematics, Statistics, Machine Learning or AI check out our YouTube Channel and LunarTech.ai for free resources.

Random Variables

Random variables form the cornerstone of many statistical concepts. It might be hard to digest the formal mathematical definition of a random variable, but simply put, it's a way to map the outcomes of random processes, such as flipping a coin or rolling a dice, to numbers.

For instance, we can define the random process of flipping a coin by random variable X which takes a value 1 if the outcome is heads and 0 if the outcome is tails.

In this example, we have a random process of flipping a coin where this experiment can produce two possible outcomes : {0,1}. This set of all possible outcomes is called the sample space of the experiment. Each time the random process is repeated, it is referred to as an event .

In this example, flipping a coin and getting a tail as an outcome is an event. The chance or the likelihood of this event occurring with a particular outcome is called the probability of that event.

A probability of an event is the likelihood that a random variable takes a specific value of x which can be described by P(x). In the example of flipping a coin, the likelihood of getting heads or tails is the same, that is 0.5 or 50%. So we have the following setting:

where the probability of an event, in this example, can only take values in the range [0,1].

To understand the concepts of mean, variance, and many other statistical topics, it is important to learn the concepts of population and sample.

The population is the set of all observations (individuals, objects, events, or procedures) and is usually very large and diverse. On the other hand, a sample is a subset of observations from the population that ideally is a true representation of the population.

Given that experimenting with an entire population is either impossible or simply too expensive, researchers or analysts use samples rather than the entire population in their experiments or trials.

To make sure that the experimental results are reliable and hold for the entire population, the sample needs to be a true representation of the population. That is, the sample needs to be unbiased.

For this purpose, we can use statistical sampling techniques such as Random Sampling, Systematic Sampling, Clustered Sampling, Weighted Sampling, and Stratified Sampling.

The mean, also known as the average, is a central value of a finite set of numbers. Let’s assume a random variable X in the data has the following values:

where N is the number of observations or data points in the sample set or simply the data frequency. Then the sample mean defined by μ , which is very often used to approximate the population mean , can be expressed as follows:

The mean is also referred to as expectation which is often defined by E () or random variable with a bar on the top. For example, the expectation of random variables X and Y, that is E (X) and E (Y), respectively, can be expressed as follows:

Now that we have a solid understanding of the mean as a statistical measure, let's see how we can apply this knowledge practically using Python. Python is a versatile programming language that, with the help of libraries like NumPy, makes it easy to perform complex mathematical operations—including calculating the mean.

In the following code snippet, we demonstrate how to compute the mean of a set of numbers using NumPy. We will start by showing the calculation for a simple array of numbers. Then, we'll address a common scenario encountered in data science: calculating the mean of a dataset that includes undefined or missing values, represented as NaN (Not a Number). NumPy provides a function specifically designed to handle such cases, allowing us to compute the mean while ignoring these NaN values.

Here is how you can perform these operations in Python:

The variance measures how far the data points are spread out from the average value. It's equal to the sum of the squares of the differences between the data values and the average (the mean).

We can express the population variance as follows:

For deriving expectations and variances of different popular probability distribution functions, check out this Github repo .

Standard Deviation

The standard deviation is simply the square root of the variance and measures the extent to which data varies from its mean. The standard deviation defined by sigma can be expressed as follows:

Standard deviation is often preferred over the variance because it has the same units as the data points, which means you can interpret it more easily.

To compute the population variance using Python, we utilize the var function from the NumPy library. By default, this function calculates the population variance by setting the ddof (Delta Degrees of Freedom) parameter to 0. However, when dealing with samples and not the entire population, you would typically set ddof to 1 to get the sample variance.

The code snippet provided shows how to calculate the variance for a set of data. Additionally, it shows how to calculate the variance when there are NaN values in the data. NaN values represent missing or undefined data. When calculating the variance, these NaN values must be handled correctly; otherwise, they can result in a variance that is not a number (NaN), which is uninformative.

Here is how you can calculate the population variance in Python, taking into account the potential presence of NaN values:

The covariance is a measure of the joint variability of two random variables and describes the relationship between these two variables. It is defined as the expected value of the product of the two random variables’ deviations from their means.

The covariance between two random variables X and Z can be described by the following expression, where E (X) and E (Z) represent the means of X and Z, respectively.

Covariance can take negative or positive values as well as a value of 0. A positive value of covariance indicates that two random variables tend to vary in the same direction, whereas a negative value suggests that these variables vary in opposite directions. Finally, the value 0 means that they don’t vary together.

To explore the concept of covariance practically, we will use Python with the NumPy library, which provides powerful numerical operations. The np.cov function can be used to calculate the covariance matrix for two or more datasets. In the matrix, the diagonal elements represent the variance of each dataset, and the off-diagonal elements represent the covariance between each pair of datasets.

Let's look at an example of calculating the covariance between two sets of data:

Correlation

The correlation is also a measure of a relationship. It measures both the strength and the direction of the linear relationship between two variables.

If a correlation is detected, then it means that there is a relationship or a pattern between the values of two target variables. Correlation between two random variables X and Z is equal to the covariance between these two variables divided by the product of the standard deviations of these variables. This can be described by the following expression:

Correlation coefficients’ values range between -1 and 1. Keep in mind that the correlation of a variable with itself is always 1, that is Cor(X, X) = 1 .

Another thing to keep in mind when interpreting correlation is to not confuse it with causation , given that a correlation is not necessarily a causation. Even if there is a correlation between two variables, you cannot conclude that one variable causes a change in the other. This relationship could be coincidental, or a third factor might be causing both variables to change.

Unit-2-Module-1---Introduction-to-Generative-AI-5

Probability Distribution Functions

A function that describes all the possible values, the sample space, and the corresponding probabilities that a random variable can take within a given range, bounded between the minimum and maximum possible values, is called a probability distribution function (pdf) or probability density.

Every pdf needs to satisfy the following two criteria:

where the first criterium states that all probabilities should be numbers in the range of [0,1] and the second criterium states that the sum of all possible probabilities should be equal to 1.

Probability functions are usually classified into two categories: discrete and continuous .

Discrete distribution function describes the random process with countable sample space, like in an example of tossing a coin that has only two possible outcomes. Continuous distribution functions describe the random process with a continuous sample space.

Examples of discrete distribution functions are Bernoulli , Binomial , Poisson , Discrete Uniform . Examples of continuous distribution functions are Normal , Continuous Uniform , Cauchy .

Binomial Distribution

The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each with the boolean-valued outcome: success (with probability p ) or failure (with probability q = 1 − p).

Let's assume a random variable X follows a Binomial distribution, then the probability of observing k successes in n independent trials can be expressed by the following probability density function:

$$\Pr(X = k) = \binom{n}{k} p^k q^{n-k}$$

The binomial distribution is useful when analyzing the results of repeated independent experiments, especially if you're interested in the probability of meeting a particular threshold given a specific error rate.

Binomial Distribution Mean and Variance

The mean of a binomial distribution, denoted as E ( X )= np , tells you the average number of successes you can expect if you conduct n independent trials of a binary experiment.

A binary experiment is one where there are only two outcomes: success (with probability p ) or failure (with probability q =1− p ).

For example, if you were to flip a coin 100 times and you define a success as the coin landing on heads (let's say the probability of heads is 0.5), the binomial distribution would tell you how likely it is to get any number of heads in those 100 flips. The mean E ( X ) would be 100×0.5=50, indicating that on average, you’d expect to get 50 heads.

The variance Var(X)=npq measures the spread of the distribution, indicating how much the number of successes is likely to deviate from the mean.

Continuing with the coin flip example, the variance would be 100×0.5×0.5=25, which informs you about the variability of the outcomes. A smaller variance would mean the results are more tightly clustered around the mean, whereas a larger variance indicates they’re more spread out.

These concepts are crucial in many fields. For instance:

Quality Control : Manufacturers might use the binomial distribution to predict the number of defective items in a batch, helping them understand the quality and consistency of their production process.
Healthcare : In medicine, it could be used to calculate the probability of a certain number of patients responding to a treatment, based on past success rates.
Finance : In finance, binomial models are used to evaluate the risk of portfolio or investment strategies by predicting the number of times an asset will reach a certain price point.
Polling and Survey Analysis : When predicting election results or customer preferences, pollsters might use the binomial distribution to estimate how many people will favor a candidate or a product, given the probability drawn from a sample.

Understanding the mean and variance of the binomial distribution is fundamental to interpreting the results and making informed decisions based on the likelihood of different outcomes.

The figure below visualizes an example of Binomial distribution where the number of independent trials is equal to 8 and the probability of success in each trial is equal to 16%.

The Python code below creates a histogram to visualize the distribution of outcomes from 1000 experiments, each consisting of 8 trials with a success probability of 0.16. It uses NumPy to generate the binomial distribution data and Matplotlib to plot the histogram, showing the probability of the number of successes in those trials.

Poisson Distribution

The Poisson distribution is the discrete probability distribution of the number of events occurring in a specified time period, given the average number of times the event occurs over that time period.

Let's assume a random variable X follows a Poisson distribution. Then the probability of observing k events over a time period can be expressed by the following probability function:

where e is Euler’s number and λ lambda, the arrival rate parameter , is the expected value of X. The Poisson distribution function is very popular for its usage in modeling countable events occurring within a given time interval.

Poisson Distribution Mean and Variance

The Poisson distribution is particularly useful for modeling the number of times an event occurs within a specified time frame. The mean E(X) and variance Var(X)

Var(X) of a Poisson distribution are both equal to λ, which is the average rate at which events occur (also known as the rate parameter). This makes the Poisson distribution unique, as it is characterized by this single parameter.

The fact that the mean and variance are equal means that as we observe more events, the distribution of the number of occurrences becomes more predictable. It’s used in various fields such as business, engineering, and science for tasks like:

Predicting the number of customer arrivals at a store within an hour. Estimating the number of emails you'd receive in a day. Understanding the number of defects in a batch of materials.

So, the Poisson distribution helps in making probabilistic forecasts about the occurrence of rare or random events over intervals of time or space.

For example, Poisson distribution can be used to model the number of customers arriving in the shop between 7 and 10 pm, or the number of patients arriving in an emergency room between 11 and 12 pm.

The figure below visualizes an example of Poisson distribution where we count the number of Web visitors arriving at the website where the arrival rate, lambda, is assumed to be equal to 7 minutes.

In practical data analysis, it is often helpful to simulate the distribution of events. Below is a Python code snippet that demonstrates how to generate a series of data points that follow a Poisson distribution using NumPy. We then create a histogram using Matplotlib to visualize the distribution of the number of visitors (as an example) we might expect to see, based on our average rate λ = 7

This histogram helps in understanding the distribution's shape and variability. The most likely number of visitors is around the mean λ, but the distribution shows the probability of seeing fewer or greater numbers as well.

Normal Distribution

The Normal probability distribution is the continuous probability distribution for a real-valued random variable. Normal distribution, also called Gaussian distribution is arguably one of the most popular distribution functions that is commonly used in social and natural sciences for modeling purposes. For example, it is used to model people’s height or test scores.

Let's assume a random variable X follows a Normal distribution. Then its probability density function can be expressed as follows:

where the parameter μ (mu) is the mean of the distribution also referred to as the location parameter , parameter σ (sigma) is the standard deviation of the distribution also referred to as the scale parameter . The number π (pi) is a mathematical constant approximately equal to 3.14.

Normal Distribution Mean and Variance

The figure below visualizes an example of Normal distribution with a mean 0 ( μ = 0 ) and standard deviation of 1 ( σ = 1 ), which is referred to as Standard Normal distribution which is symmetric .

The visualization of the standard normal distribution is crucial because this distribution underpins many statistical methods and probability theory. When data is normally distributed with a mean ( μ ) of 0 and standard deviation (σ) of 1, it is referred to as the standard normal distribution. It's symmetric around the mean, with the shape of the curve often called the "bell curve" due to its bell-like shape.

The standard normal distribution is fundamental for the following reasons:

Central Limit Theorem: This theorem states that, under certain conditions, the sum of a large number of random variables will be approximately normally distributed. It allows for the use of normal probability theory for sample means and sums, even when the original data is not normally distributed.
Z-Scores: Values from any normal distribution can be transformed into the standard normal distribution using Z-scores, which indicate how many standard deviations an element is from the mean. This allows for the comparison of scores from different normal distributions.
Statistical Inference and AB Testing: Many statistical tests, such as t-tests and ANOVAs, assume that the data follows a normal distribution, or they rely on the central limit theorem. Understanding the standard normal distribution helps in the interpretation of these tests' results.
Confidence Intervals and Hypothesis Testing: The properties of the standard normal distribution are used to construct confidence intervals and to perform hypothesis testing.

All topics which we will cover below!

So, being able to visualize and understand the standard normal distribution is key to applying many statistical techniques accurately.

The Python code below uses NumPy to generate 1000 random samples from a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1, which are standard parameters for the standard normal distribution. These generated samples are stored in the variable X.

To visualize the distribution of these samples, the code employs Matplotlib to create a histogram. The plt.hist function is used to plot the histogram of the samples with 30 bins, and the density parameter is set to True to normalize the histogram so that the area under it sums to 1. This effectively turns the histogram into a probability density plot.

Additionally, the SciPy library is used to overlay the probability density function (PDF) of the theoretical normal distribution on the histogram. The norm.pdf function generates the y-values for the PDF given an array of x-values. This theoretical curve is plotted in yellow over the histogram to show how closely the random samples fit the expected distribution.

The resulting graph displays the histogram of the generated samples in purple, with the theoretical normal distribution overlaid in yellow. The x-axis represents the range of values that the samples can take, while the y-axis represents the probability density. This visualization is a powerful tool for comparing the empirical distribution of the data with the theoretical model, allowing us to see whether our samples follow the expected pattern of a normal distribution.

Unit-2-Module-1---Introduction-to-Generative-AI-7

Bayes' Theorem

The Bayes' Theorem (often called Bayes' Law ) is arguably the most powerful rule of probability and statistics. It was named after famous English statistician and philosopher, Thomas Bayes.

Bayes' theorem is a powerful probability law that brings the concept of subjectivity into the world of Statistics and Mathematics where everything is about facts. It describes the probability of an event, based on the prior information of conditions that might be related to that event.

For instance, if the risk of getting Coronavirus or Covid-19 is known to increase with age, then Bayes' Theorem allows the risk to an individual of a known age to be determined more accurately. It does this by conditioning it on the age rather than simply assuming that this individual is common to the population as a whole.

The concept of conditional probability , which plays a central role in Bayes' theorem, is a measure of the probability of an event happening, given that another event has already occurred.

Bayes' theorem can be described by the following expression where the X and Y stand for events X and Y, respectively:

Pr (X|Y): the probability of event X occurring given that event or condition Y has occurred or is true
Pr (Y|X): the probability of event Y occurring given that event or condition X has occurred or is true
Pr (X) & Pr (Y): the probabilities of observing events X and Y, respectively

In the case of the earlier example, the probability of getting Coronavirus (event X) conditional on being at a certain age is Pr (X|Y). This is equal to the probability of being at a certain age given that the person got a Coronavirus, Pr (Y|X), multiplied with the probability of getting a Coronavirus, Pr (X), divided by the probability of being at a certain age, Pr (Y).

Linear Regression

Earlier, we introduced the concept of causation between variables, which happens when a variable has a direct impact on another variable.

When the relationship between two variables is linear, then Linear Regression is a statistical method that can help model the impact of a unit change in a variable, the independent variable on the values of another variable, the dependent variable .

Dependent variables are often referred to as response variables or explained variables, whereas independent variables are often referred to as regressors or explanatory variables .

When the Linear Regression model is based on a single independent variable, then the model is called Simple Linear Regression . When the model is based on multiple independent variables, it’s referred to as Multiple Linear Regression.

Simple Linear Regression can be described by the following expression:

where Y is the dependent variable, X is the independent variable which is part of the data, β0 is the intercept which is unknown and constant, β1 is the slope coefficient or a parameter corresponding to the variable X which is unknown and constant as well. Finally, u is the error term that the model makes when estimating the Y values.

The main idea behind linear regression is to find the best-fitting straight line, the regression line, through a set of paired ( X, Y ) data.

One example of the Linear Regression application is modeling the impact of flipper length on penguins’ body mass, which is visualized below:

The R code snippet you've shared is for creating a scatter plot with a linear regression line using the ggplot2 package in R, which is a powerful and widely-used library for creating graphics and visualizations. The code uses a dataset named penguins from the palmerpenguins package, presumably containing data about penguin species, including measurements like flipper length and body mass.

Multiple Linear Regression with three independent variables can be described by the following expression:

Ordinary Least Squares

The ordinary least squares (OLS) is a method for estimating the unknown parameters such as β0 and β1 in a linear regression model. The model is based on the principle of least squares . This minimizes the sum of the squares of the differences between the observed dependent variable and its values that are predicted by the linear function of the independent variable (often referred to as fitted values ).

This difference between the real and predicted values of dependent variable Y is referred to as residual . So OLS minimizes the sum of squared residuals.

This optimization problem results in the following OLS estimates for the unknown parameters β0 and β1 which are also known as coefficient estimates :

Once these parameters of the Simple Linear Regression model are estimated, the fitted values of the response variable can be computed as follows:

Standard Error

The residuals or the estimated error terms can be determined as follows:

It is important to keep in mind the difference between the error terms and residuals. Error terms are never observed, while the residuals are calculated from the data. The OLS estimates the error terms for each observation but not the actual error term. So, the true error variance is still unknown.

Also, these estimates are subject to sampling uncertainty. This means that we will never be able to determine the exact estimate, the true value, of these parameters from sample data in an empirical application. But we can estimate it by calculating the sample residual variance by using the residuals as follows:

This estimate for the variance of sample residuals helps us estimate the variance of the estimated parameters, which is often expressed as follows:

The square root of this variance term is called the standard error of the estimate. This is a key component in assessing the accuracy of the parameter estimates. It is used to calculate test statistics and confidence intervals.

The standard error can be expressed as follows:

It is important to keep in mind the difference between the error terms and residuals. Error terms are never observed, while the residuals are calculated from the data.

OLS Assumptions

The OLS estimation method makes the following assumptions which need to be satisfied to get reliable prediction results:

The Linearity assumption states that the model is linear in parameters.
The Random Sample assumption states that all observations in the sample are randomly selected.
The Exogeneity assumption states that independent variables are uncorrelated with the error terms.
The Homoskedasticity assumption states that the variance of all error terms is constant.
The No Perfect Multi-Collinearity assumption states that none of the independent variables is constant and there are no exact linear relationships between the independent variables.

The Python code snippet you've shared performs Ordinary Least Squares (OLS) regression, which is a method used in statistics to estimate the relationship between independent variables and a dependent variable. This process involves calculating the best-fit line through the data points that minimizes the sum of the squared differences between the observed values and the values predicted by the model.

The code defines a function runOLS(Y, X) that takes in a dependent variable Y and an independent variable X and performs the following steps:

Estimates the OLS coefficients (beta_hat) using the linear algebra solution to the least squares problem.
Makes predictions ( Y_hat ) using the estimated coefficients and calculates the residuals.
Computes the residual sum of squares (RSS), total sum of squares (TSS), mean squared error (MSE), root mean squared error (RMSE), and R-squared value, which are common metrics used to assess the fit of the model.
Calculates the standard error of the coefficient estimates, t-statistics, p-values, and confidence intervals for the estimated coefficients.

These calculations are standard in regression analysis and are used to interpret and understand the strength and significance of the relationship between the variables. The result of this function includes the estimated coefficients and various statistics that help evaluate the model's performance.

Parameter Properties

Under the assumption that the OLS criteria/assumptions we just discussed are satisfied, the OLS estimators of coefficients β0 and β1 are BLUE and Consistent . So what does this mean?

This theorem highlights the properties of OLS estimates where the term BLUE stands for Best Linear Unbiased Estimator . Let's explore what this means in more detail.

The bias of an estimator is the difference between its expected value and the true value of the parameter being estimated. It can be expressed as follows:

When we state that the estimator is unbiased , we mean that the bias is equal to zero. This implies that the expected value of the estimator is equal to the true parameter value, that is:

Unbiasedness does not guarantee that the obtained estimate with any particular sample is equal or close to β. What it means is that, if we repeatedly draw random samples from the population and then computes the estimate each time, then the average of these estimates would be equal or very close to β.

The term Best in the Gauss-Markov theorem relates to the variance of the estimator and is referred to as efficiency . A parameter can have multiple estimators but the one with the lowest variance is called efficient.

Consistency

The term consistency goes hand in hand with the terms sample size and convergence . If the estimator converges to the true parameter as the sample size becomes very large, then this estimator is said to be consistent, that is:

All these properties hold for OLS estimates as summarized in the Gauss-Markov theorem. In other words, OLS estimates have the smallest variance, they are unbiased, linear in parameters, and are consistent. These properties can be mathematically proven by using the OLS assumptions made earlier.

Confidence Intervals

The Confidence Interval is the range that contains the true population parameter with a certain pre-specified probability. This is referred to as the confidence level of the experiment, and it's obtained by using the sample results and the margin of error .

Margin of Error

The margin of error is the difference between the sample results and based on what the result would have been if you had used the entire population.

Confidence Level

The Confidence Level describes the level of certainty in the experimental results. For example, a 95% confidence level means that if you were to perform the same experiment repeatedly 100 times, then 95 of those 100 trials would lead to similar results.

Note that the confidence level is defined before the start of the experiment because it will affect how big the margin of error will be at the end of the experiment.

Confidence Interval for OLS Estimates

As I mentioned earlier, the OLS estimates of the Simple Linear Regression, the estimates for intercept β0 and slope coefficient β1, are subject to sampling uncertainty. But we can construct Confidence Intervals (CIs) for these parameters which will contain the true value of these parameters in 95% of all samples.

That is, 95% confidence interval for β can be interpreted as follows:

The confidence interval is the set of values for which a hypothesis test cannot be rejected to the level of 5%.
The confidence interval has a 95% chance to contain the true value of β.

95% confidence interval of OLS estimates can be constructed as follows:

This is based on the parameter estimate, the standard error of that estimate, and the value 1.96 representing the margin of error corresponding to the 5% rejection rule.

This value is determined using the Normal Distribution table , which we'll discuss later on in this handbook.

Meanwhile, the following figure illustrates the idea of 95% CI:

Note that the confidence interval depends on the sample size as well, given that it is calculated using the standard error which is based on sample size.

Statistical Hypothesis Testing

Testing a hypothesis in Statistics is a way to test the results of an experiment or survey to determine how meaningful they the results are.

Basically, you're testing whether the obtained results are valid by figuring out the odds that the results have occurred by chance. If it is the letter, then the results are not reliable and neither is the experiment. Hypothesis Testing is part of the Statistical Inference .

Null and Alternative Hypothesis

Firstly, you need to determine the thesis you wish to test. Then you need to formulate the Null Hypothesis and the Alternative Hypothesis. The test can have two possible outcomes. Based on the statistical results, you can either reject the stated hypothesis or accept it.

As a rule of thumb, statisticians tend to put the version or formulation of the hypothesis under the Null Hypothesis that needs to be rejected , whereas the acceptable and desired version is stated under the Alternative Hypothesis .

Statistical Significance

Let’s look at the earlier mentioned example where we used the Linear Regression model to investigate whether a penguin's Flipper Length, the independent variable, has an impact on Body Mass , the dependent variable.

We can formulate this model with the following statistical expression:

Then, once the OLS estimates of the coefficients are estimated, we can formulate the following Null and Alternative Hypothesis to test whether the Flipper Length has a statistically significant impact on the Body Mass:

where H0 and H1 represent Null Hypothesis and Alternative Hypothesis, respectively.

Rejecting the Null Hypothesis would mean that a one-unit increase in Flipper Length has a direct impact on the Body Mass (given that the parameter estimate of β1 is describing this impact of the independent variable, Flipper Length, on the dependent variable, Body Mass). We can reformulate this hypothesis as follows:

where H0 states that the parameter estimate of β1 is equal to 0, that is Flipper Length effect on Body Mass is statistically insignificant whereas H1 states that the parameter estimate of β1 is not equal to 0, suggesting that Flipper Length effect on Body Mass is statistically significant .

Type I and Type II Errors

When performing Statistical Hypothesis Testing, you need to consider two conceptual types of errors: Type I error and Type II error.

Type I errors occur when the Null is incorrectly rejected, and Type II errors occur when the Null Hypothesis is incorrectly not rejected. A confusion matrix can help you clearly visualize the severity of these two types of errors.

As a rule of thumb, statisticians tend to put the version of the hypothesis under the Null Hypothesis that that needs to be rejected, whereas the acceptable and desired version is stated under the Alternative Hypothesis.

Unit-2-Module-1---Introduction-to-Generative-AI-3-1

Statistical Tests

Once the you've stataed the Null and the Alternative Hypotheses and defined the test assumptions, the next step is to determine which statistical test is appropriate and to calculate the test statistic .

Whether or not to reject or not reject the Null can be determined by comparing the test statistic with the critical value . This comparison shows whether or not the observed test statistic is more extreme than the defined critical value.

It can have two possible results:

The test statistic is more extreme than the critical value → the null hypothesis can be rejected
The test statistic is not as extreme as the critical value → the null hypothesis cannot be rejected

The critical value is based on a pre-specified significance level α (usually chosen to be equal to 5%) and the type of probability distribution the test statistic follows.

The critical value divides the area under this probability distribution curve into the rejection region(s) and non-rejection region . There are numerous statistical tests used to test various hypotheses. Examples of Statistical tests are Student’s t-test , F-test , Chi-squared test , Durbin-Hausman-Wu Endogeneity test , W hite Heteroskedasticity test . In this handbook, we will look at two of these statistical tests: the Student's t-test and the F-test.

Student’s t-test

One of the simplest and most popular statistical tests is the Student’s t-test. You can use it to test various hypotheses, especially when dealing with a hypothesis where the main area of interest is to find evidence for the statistically significant effect of a single variable .

The test statistics of the t-test follows Student’s t distribution and can be determined as follows:

where h0 in the nominator is the value against which the parameter estimate is being tested. So, the t-test statistics are equal to the parameter estimate minus the hypothesized value divided by the standard error of the coefficient estimate.

Let's use this for our earlier hypothesis, where we wanted to test whether Flipper Length has a statistically significant impact on Body Mass or not. This test can be performed using a t-test. The h0 is in that case equal to the 0 since the slope coefficient estimate is tested against a value of 0.

Two-sided vs one-sided t-test

There are two versions of the t-test: a two-sided t-test and a one-sided t-test . Whether you need the former or the latter version of the test depends entirely on the hypothesis that you want to test.

You can use the two-sided or two-tailed t-test when the hypothesis is testing equal versus not equal relationship under the Null and Alternative Hypotheses. It would be similar to the following example:

The two-sided t-test has two rejection regions as visualized in the figure below:

In this version of the t-test, the Null is rejected if the calculated t-statistics is either too small or too large.

Here, the test statistics are compared to the critical values based on the sample size and the chosen significance level. To determine the exact value of the cutoff point, you can use a two-sided t-distribution table .

On the other hand, you can use the one-sided or one-tailed t-test when the hypothesis is testing positive/negative versus negative/positive relationships under the Null and Alternative Hypotheses. It looks like this:

One-sided t-test has a single rejection region . Depending on the hypothesis side, the rejection region is either on the left-hand side or the right-hand side as visualized in the figure below:

In this version of the t-test, the Null is rejected if the calculated t-statistics is smaller/larger than the critical value.

F-test is another very popular statistical test often used to test hypotheses testing a joint statistical significance of multiple variables . This is the case when you want to test whether multiple independent variables have a statistically significant impact on a dependent variable.

Following is an example of a statistical hypothesis that you can test using the F-test:

where the Null states that the three variables corresponding to these coefficients are jointly statistically insignificant, and the Alternative states that these three variables are jointly statistically significant.

The test statistics of the F-test follows F distribution and can be determined as follows:

the SSRrestricted is the sum of squared residuals of the restricted model , which is the same model excluding from the data the target variables stated as insignificant under the Null
the SSRunrestricted is the sum of squared residuals of the unrestricted model , which is the model that includes all variables
the q represents the number of variables that are being jointly tested for the insignificance under the Null
N is the sample size
and the k is the total number of variables in the unrestricted model.

SSR values are provided next to the parameter estimates after running the OLS regression, and the same holds for the F-statistics as well.

Following is an example of MLR model output where the SSR and F-statistics values are marked.

F-test has a single rejection region as visualized below:

If the calculated F-statistics is bigger than the critical value, then the Null can be rejected. This suggests that the independent variables are jointly statistically significant. The rejection rule can be expressed as follows:

2-sample T-test

If you want to test whether there is a statistically significant difference between the control and experimental groups’ metrics that are in the form of averages (for example, average purchase amount), metric follows student-t distribution. When the sample size is smaller than 30, you can use 2-sample T-test to test the following hypothesis:

where the sampling distribution of means of Control group follows Student-t distribution with degrees of freedom N_con-1. Also, the sampling distribution of means of the Experimental group also follows the Student-t distribution with degrees of freedom N_exp-1.

Note that the N_con and N_exp are the number of users in the Control and Experimental groups, respectively.

Then you can calculate an estimate for the pooled variance of the two samples as follows:

where σ²_con and σ²_exp are the sample variances of the Control and Experimental groups, respectively. Then the Standard Error is equal to the square root of the estimate of the pooled variance, and can be defined as:

Consequently, the test statistics of the 2-sample T-test with the hypothesis stated earlier can be calculated as follows:

In order to test the statistical significance of the observed difference between sample means, we need to calculate the p-value of our test statistics.

The p-value is the probability of observing values at least as extreme as the common value when this is due to a random chance. Stated differently, the p-value is the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the null hypothesis is true.

Then the p-value of the test statistics can be calculated as follows:

The interpretation of a p -value is dependent on the chosen significance level, alpha, which you choose before running the test during the power analysis .

If the calculated p -value appears to be smaller than equal to alpha (for example, 0.05 for 5% significance level) we can reject the null hypothesis and state that there is a statistically significant difference between the primary metrics of the Control and Experimental groups.

Finally, to determine how accurate the obtained results are and also to comment about the practical significance of the obtained results, you can compute the Confidence Interval of your test by using the following formula:

where the t_(1-alpha/2) is the critical value of the test corresponding to the two-sided t-test with alpha significance level. It can be found using the t-table .

The Python code provided performs a two-sample t-test, which is used in statistics to determine if two sets of data are significantly different from each other. This particular snippet simulates two groups (control and experimental) with data following a t-distribution, calculates the mean and variance for each group, and then performs the following steps:

It calculates the pooled variance, which is a weighted average of the variances of the two groups.
It computes the standard error of the difference between the two means.
It calculates the t-statistic, which is the difference between the two sample means divided by the standard error. This statistic measures how much the groups differ in units of standard error.
It determines the critical t-value from the t-distribution for the given significance level and degrees of freedom, which is used to decide whether the t-statistic is large enough to indicate a statistically significant difference between the groups.
It calculates the p-value, which indicates the probability of observing such a difference between means if the null hypothesis (that there is no difference) is true.
It computes the margin of error and constructs the confidence interval around the difference in means.

Finally, the code prints out the t-statistic, critical t-value, p-value, and confidence interval. These results can be used to infer whether the observed differences in means are statistically significant or likely due to random variation.

2-sample Z-test

There are various situations when you may want to use a 2-sample z-test:

if you want to test whether there is a statistically significant difference between the control and experimental groups’ metrics that are in the form of averages (for example, average purchase amount) or proportions (for example, Click Through Rate)
if the metric follows Normal distribution
when the sample size is larger than 30, such that you can use the Central Limit Theorem (CLT) to state that the sampling distributions of the Control and Experimental groups are asymptotically Normal

Here we will make a distinction between two cases: where the primary metric is in the form of proportions (like Click Through Rate) and where the primary metric is in the form of averages (like average purchase amount).

Case 1: Z-test for comparing proportions (2-sided)

If you want to test whether there is a statistically significant difference between the Control and Experimental groups’ metrics that are in the form of proportions (like CTR) and if the click event occurs independently, you can use a 2-sample Z-test to test the following hypothesis:

where each click event can be described by a random variable that can take two possible values: 1 (success) and 0 (failure). It also follows a Bernoulli distribution (click: success and no click: failure) where p_con and p_exp are the probabilities of clicking (probability of success) of Control and Experimental groups, respectively.

So, after collecting the interaction data of the Control and Experimental users, you can calculate the estimates of these two probabilities as follows:

Since we are testing for the difference in these probabilities, we need to obtain an estimate for the pooled probability of success and an estimate for pooled variance, which can be done as follows:

Then the Standard Error is equal to the square root of the estimate of the pooled variance. It can be defined as:

And so, the test statistics of the 2-sample Z-test for the difference in proportions can be calculated as follows:

Then the p-value of this test statistics can be calculated as follows:

Finally, you can compute the Confidence Interval of the test as follows:

where the z_(1-alpha/2) is the critical value of the test corresponding to the two-sided Z-test with alpha significance level. You can find it using the Z-table .

The rejection region of this two-sided 2-sample Z-test can be visualized by the following graph:

The Python code snippet you’ve provided performs a two-sample Z-test for proportions. This type of test is used to determine whether there is a significant difference between the proportions of two groups. Here’s a brief explanation of the steps the code performs:

Calculates the sample proportions for both the control and experimental groups.
Computes the pooled sample proportion, which is an estimate of the proportion assuming the null hypothesis (that there is no difference between the group proportions) is true.
Calculates the pooled sample variance based on the pooled proportion and the sizes of the two samples.
Derives the standard error of the difference in sample proportions.
Calculates the Z-test statistic, which measures the number of standard errors between the sample proportion difference and the null hypothesis.
Finds the critical Z-value from the standard normal distribution for the given significance level.
Computes the p-value to assess the evidence against the null hypothesis.
Calculates the margin of error and the confidence interval for the difference in proportions.
Outputs the test statistic, critical value, p-value, and confidence interval, and based on the test statistic and critical value, it may print a statement to either reject or not reject the null hypothesis.

The latter part of the code uses Matplotlib to create a visualization of the standard normal distribution and the rejection regions for the two-sided Z-test. This visual aid helps to understand where the test statistic falls in relation to the distribution and the critical values.

Case 2: Z-test for Comparing Means (2-sided)

If you want to test whether there is a statistically significant difference between the Control and Experimental groups’ metrics that are in the form of averages (like average purchase amount) you can use a 2-sample Z-test to test the following hypothesis:

where the sampling distribution of means of the Control group follows Normal distribution with mean mu_con and σ²_con/N_con. Moreover, the sampling distribution of means of the Experimental group also follows the Normal distribution with mean mu_exp and σ²_exp/N_exp.

Then the difference in the means of the control and experimental groups also follows Normal distributions with mean mu_con-mu_exp and variance σ²_con/N_con + σ²_exp/N_exp.

Consequently, the test statistics of the 2-sample Z-test for the difference in means can be calculated as follows:

The Standard Error is equal to the square root of the estimate of the pooled variance and can be defined as:

The Python code provided appears to be set up for conducting a two-sample Z-test, typically used to determine if there is a significant difference between the means of two independent groups. In this context, the code might be comparing two different processes or treatments.

It generates two arrays of random integers to represent data for a control group ( X_A ) and an experimental group ( X_B ).
It calculates the sample means ( mu_con , mu_exp ) and variances ( variance_con , variance_exp ) for both groups.
The pooled variance is computed, which is used in the denominator of the test statistic formula for the Z-test, providing a measure of the data's common variance.
The Z-test statistic ( T ) is calculated by taking the difference between the two sample means and dividing it by the standard error of the difference.
The p-value is calculated to test the hypothesis of whether the means of the two groups are statistically different from each other.
The critical Z-value ( Z_crit ) is determined from the standard normal distribution, which defines the cutoff points for significance.
A margin of error is computed, and a confidence interval for the difference in means is constructed.
The test statistic, critical value, p-value, and confidence interval are printed to the console.

Lastly, the code uses Matplotlib to plot the standard normal distribution and highlight the rejection regions for the Z-test. This visualization can help in understanding the result of the Z-test in terms of where the test statistic lies relative to the distribution and the critical values for a two-sided test.

Chi-Squared test

If you want to test whether there is a statistically significant difference between the Control and Experimental groups’ performance metrics (for example their conversions) and you don’t really want to know the nature of this relationship (which one is better) you can use a Chi-Squared test to test the following hypothesis:

Note that the metric should be in the form of a binary variable (for example, conversion or no conversion/click or no click). The data can then be represented in the form of the following table, where O and T correspond to observed and theoretical values, respectively.

Then the test statistics of the Chi-2 test can be expressed as follows:

where the Observed corresponds to the observed data and the Expected corresponds to the theoretical value, and i can take values 0 (no conversion) and 1(conversion). It’s important to see that each of these factors has a separate denominator. The formula for the test statistics when you have two groups only can be represented as follows:

The expected value is simply equal to the number of times each version of the product is viewed multiplied by the probability of it leading to conversion (or to a click in case of CTR).

Note that, since the Chi-2 test is not a parametric test, its Standard Error and Confidence Interval can’t be calculated in a standard way as we did in the parametric Z-test or T-test.

The Python code you've shared is for conducting a Chi-squared test, a statistical hypothesis test that is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.

In the provided code snippet, it looks like the test is being used to compare two categorical datasets:

It calculates the Chi-squared test statistic by summing the squared difference between observed ( O ) and expected ( T ) frequencies, divided by the expected frequencies for each category. This is known as the squared relative distance and is used as the test statistic for the Chi-squared test.
It then calculates the p-value for this test statistic using the degrees of freedom, which in this case is assumed to be 1 (but this would typically depend on the number of categories minus one).
The Matplotlib library is used to plot the probability density function (pdf) of the Chi-squared distribution with one degree of freedom. It also highlights the rejection region for the test, which corresponds to the critical value of the Chi-squared distribution that the test statistic must exceed for the difference to be considered statistically significant.

The visualization helps to understand the Chi-squared test by showing where the test statistic lies in relation to the Chi-squared distribution and its critical value. If the test statistic is within the rejection region, the null hypothesis of no difference in frequencies can be rejected.

Another quick way to determine whether to reject or to support the Null Hypothesis is by using p-values . The p-value is the probability of the condition under the Null occurring. Stated differently, the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic. The smaller the p-value, the stronger is the evidence against the Null Hypothesis, suggesting that it can be rejected.

The interpretation of a p -value is dependent on the chosen significance level. Most often, 1%, 5%, or 10% significance levels are used to interpret the p-value. So, instead of using the t-test and the F-test, p-values of these test statistics can be used to test the same hypotheses.

The following figure shows a sample output of an OLS regression with two independent variables. In this table, the p-value of the t-test, testing the statistical significance of class_size variable’s parameter estimate, and the p-value of the F-test, testing the joint statistical significance of the class_size, and el_pct variables parameter estimates, are underlined.

The p-value corresponding to the class_size variable is 0.011. When we compare this value to the significance levels 1% or 0.01 , 5% or 0.05, 10% or 0.1, then we can make the following conclusions:

0.011 > 0.01 → Null of the t-test can’t be rejected at 1% significance level
0.011 < 0.05 → Null of the t-test can be rejected at 5% significance level
0.011 < 0.10 → Null of the t-test can be rejected at 10% significance level

So, this p-value suggests that the coefficient of the class_size variable is statistically significant at 5% and 10% significance levels. The p-value corresponding to the F-test is 0.0000. And since 0 is smaller than all three cutoff values (0.01, 0.05, 0.10), we can conclude that the Null of the F-test can be rejected in all three cases.

This suggests that the coefficients of class_size and el_pct variables are jointly statistically significant at 1%, 5%, and 10% significance levels.

Limitation of p-values

Using p-values has many benefits, but it has also limitations. One of the main ones is that the p-value depends on both the magnitude of association and the sample size. If the magnitude of the effect is small and statistically insignificant, the p-value might still show a significant impact because the sample size is large. The opposite can occur as well – an effect can be large, but fail to meet the p<0.01, 0.05, or 0.10 criteria if the sample size is small.

Inferential statistics uses sample data to make reasonable judgments about the population from which the sample data originated. We use it to investigate the relationships between variables within a sample and make predictions about how these variables will relate to a larger population.

Both the Law of Large Numbers (LLN) and the Central Limit Theorem (CLM) have a significant role in Inferential statistics because they show that the experimental results hold regardless of what shape the original population distribution was when the data is large enough.

The more data is gathered, the more accurate the statistical inferences become – hence, the more accurate parameter estimates are generated.

Law of Large Numbers (LLN)

Suppose X1, X2, . . . , Xn are all independent random variables with the same underlying distribution (also called independent identically-distributed or i.i.d), where all X’s have the same mean μ and standard deviation σ . As the sample size grows, the probability that the average of all X’s is equal to the mean μ is equal to 1.

The Law of Large Numbers can be summarized as follows:

Central Limit Theorem (CLM)

Suppose X1, X2, . . . , Xn are all independent random variables with the same underlying distribution (also called independent identically-distributed or i.i.d), where all X’s have the same mean μ and standard deviation σ . As the sample size grows, the probability distribution of X converges in the distribution in Normal distribution with mean μ and variance σ- squared.

The Central Limit Theorem can be summarized as follows:

Stated differently, when you have a population with mean μ and standard deviation σ and you take sufficiently large random samples from that population with replacement, then the distribution of the sample means will be approximately normally distributed.

Dimensionality Reduction Techniques

Dimensionality reduction is the transformation of data from a high-dimensional space into a low-dimensional space such that this low-dimensional representation of the data still contains the meaningful properties of the original data as much as possible.

With the increase in popularity in Big Data, the demand for these dimensionality reduction techniques, reducing the amount of unnecessary data and features, increased as well. Examples of popular dimensionality reduction techniques are Principle Component Analysis , Factor Analysis , Canonical Correlation , Random Forest .

Principle Component Analysis (PCA)

Principal Component Analysis (PCA) is a dimensionality reduction technique that is very often used to reduce the dimensionality of large data sets. It does this by transforming a large set of variables into a smaller set that still contains most of the information or the variation in the original large dataset.

Let’s assume we have a data X with p variables X1, X2, …., Xp with eigenvectors e1, …, ep, and eigenvalues λ1,…, λp. Eigenvalues show the variance explained by a particular data field out of the total variance.

The idea behind PCA is to create new (independent) variables, called Principal Components, that are a linear combination of the existing variable. The i th principal component can be expressed as follows:

Then using the Elbow Rule or Kaiser Rule , you can determine the number of principal components that optimally summarize the data without losing too much information.

It is also important to look at the proportion of total variation (PRTV) that is explained by each principal component to decide whether it is beneficial to include or to exclude it. PRTV for the i th principal component can be calculated using eigenvalues as follows:

The elbow rule or the elbow method is a heuristic approach that we can use to determine the number of optimal principal components from the PCA results.

The idea behind this method is to plot the explained variation as a function of the number of components and pick the elbow of the curve as the number of optimal principal components.

Following is an example of such a scatter plot where the PRTV (Y-axis) is plotted on the number of principal components (X-axis). The elbow corresponds to the X-axis value 2, which suggests that the number of optimal principal components is 2.

Factor Analysis (FA)

Factor analysis or FA is another statistical method for dimensionality reduction. It is one of the most commonly used inter-dependency techniques. We can use it when the relevant set of variables shows a systematic inter-dependence and our objective is to find out the latent factors that create a commonality.

Let’s assume we have a data X with p variables X1, X2, …., Xp. The FA model can be expressed as follows:

X is a [p x N] matrix of p variables and N observations
µ is [p x N] population mean matrix
A is [p x k] common factor loadings matrix
F [k x N] is the matrix of common factors
and u [pxN] is the matrix of specific factors.

So, to put it differently, a factor model is as a series of multiple regressions, predicting each of the variables Xi from the values of the unobservable common factors are:

Each variable has k of its own common factors, and these are related to the observations via the factor loading matrix for a single observation as follows:

In factor analysis, the factors are calculated to maximize between-group variance while minimizing in-group varianc e. They are factors because they group the underlying variables. Unlike the PCA, in FA the data needs to be normalized, given that FA assumes that the dataset follows Normal Distribution.

Interview Prep – Top 7 Statistics Questions with Answers

Are you preparing for interviews in statistics, data analysis, or data science? It's crucial to know key statistical concepts and their applications.

Below I've included seven important statistics questions with answers, covering basic statistical tests, probability theory, and the use of statistics in decision-making, like A/B testing.

Question 1: What is the d ifference b etween a t-test and Z-test ?

The question "What is the difference between a t-test and Z-test?" is a common question in data science interviews because it tests the candidate's understanding of basic statistical concepts used in comparing group means.

This knowledge is crucial because choosing the right test affects the validity of conclusions drawn from data, which is a daily task in a data scientist's role when it comes to interpreting experiments, analyzing survey results, or evaluating models.

Both t-tests and Z-tests are statistical methods used to determine if there are significant differences between the means of two groups. But they have key differences:

Assumptions : You can use a t-test when the sample sizes are small and the population standard deviation is unknown. It doesn't require the sample mean to be normally distributed if the sample size is sufficiently large due to the Central Limit Theorem. The Z-test assumes that both the sample and the population distributions are normally distributed.
Sample Size : T-tests are typically used for sample sizes smaller than 30, whereas Z-tests are used for larger sample sizes (greater than or equal to 30) when the population standard deviation is known.
Test Statistic : The t-test uses the t-distribution to calculate the test statistic, taking into account the sample standard deviation. The Z-test uses the standard normal distribution, utilizing the known population standard deviation.
P-Value : The p-value in a t-test is determined based on the t-distribution, which accounts for the variability in smaller samples. The Z-test uses the standard normal distribution to calculate the p-value, suitable for larger samples or known population parameters.

Question 2: What is a p-value?

The question "What is a p-value?" requires the understanding of a fundamental concept in hypothesis testing that we descussed in this blog in detail with examples. It's not just a number – it's a bridge between the data you collect and the conclusions you draw for data driven decision making.

P-values quantify the evidence against a null hypothesis—how likely it is to observe the collected data if the null hypothesis were true.

For data scientists, p-values are part of everyday language in statistical analysis, model validation, and experimental design. They have to interpret p-values correctly to make informed decisions and often need to explain their implications to stakeholders who might not have deep statistical knowledge.

Thus, understanding p-values helps data scientists to convey the level of certainty or doubt in their findings and to justify subsequent actions or recommendations.

So here you need to show your understanding of what p-value measures and connect it to statistical significance and hypothesis testing.

The p-value measures the probability of observing a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. It helps in deciding whether the observed data significantly deviate from what would be expected under the null hypothesis.

If the p-value is lower than a predetermined threshold (alpha level, usually set at 0.05), the null hypothesis is rejected, indicating that the observed result is statistically significant.

Question 3: What are limitations of p-values?

P-values are a staple of inferential statistics, providing a metric for evaluating evidence against a null hypothesis. In these question you need to name couple of them.

Dependence on Sample Size : The p-value is sensitive to the sample size. Large samples might yield significant p-values even for trivial effects, while small samples may not detect significant effects even if they exist.
Not a Measure of Effect Size or Importance : A small p-value does not necessarily mean the effect is practically significant – it simply indicates it's unlikely to have occurred by chance.
Misinterpretation : P-values can be misinterpreted as the probability that the null hypothesis is true, which is incorrect. They only measure the evidence against the null hypothesis.

Question 4: What is a Confidence Level?

A confidence level represents the frequency with which an estimated confidence interval would contain the true population parameter if the same process were repeated multiple times.

For example, a 95% confidence level means that if the study were repeated 100 times, approximately 95 of the confidence intervals calculated from those studies would be expected to contain the true population parameter.

Question 5: What is the Probability of Picking 5 Red and 5 Blue Balls Without Replacement?

What is the probability of picking exactly 5 red balls and 5 blue balls in 10 picks without replacement from a set of 100 balls, where there are 70 red balls and 30 blue balls? The text describes how to calculate this probability using combinatorial mathematics and the hypergeometric distribution.

In this question, you're dealing with a classic probability problem that involves combinatorial principles and the concept of probability without replacement. The context is a finite set of balls, each draw affecting the subsequent ones because the composition of the set changes with each draw.

To approach this problem, you need to consider:

The total number of balls : If the question doesn't specify this, you need to ask or make a reasonable assumption based on the context.
Initial proportion of balls : Know the initial count of red and blue balls in the set.
Sequential probability : Remember that each time you draw a ball, you don't put it back, so the probability of drawing a ball of a certain color changes with each draw.
Combinations : Calculate the number of ways to choose 5 red balls from the total red balls and 5 blue balls from the total blue balls, then divide by the number of ways to choose any 10 balls from the total.

Thinking through these points will guide you in formulating the solution based on the hypergeometric distribution, which describes the probability of a given number of successes in draws without replacement from a finite population.

This question tests your ability to apply probability theory to a dynamic scenario, a skill that's invaluable in data-driven decision-making and statistical modeling.

To find the probability of picking exactly 5 red balls and 5 blue balls in 10 picks without replacement, we calculate the probability of picking 5 red balls out of 70 and 5 blue balls out of 30, and then divide by the total ways to pick 10 balls out of 100:

Let's calculate this probability:

Question 6: Explain Bayes' Theorem and its importance in calculating posterior probabilities.

Provide an example of how it might be used in genetic testing to determine the likelihood of an individual carrying a certain gene.

Bayes' Theorem is a cornerstone of probability theory that enables the updating of initial beliefs (prior probabilities) with new evidence to obtain updated beliefs (posterior probabilities). This question wants to test candidates ability to explain the concept, mathematical framework for incorporating new evidence into existing predictions or models.

Bayes' Theorem is a fundamental theorem in probability theory and statistics that describes the probability of an event, based on prior knowledge of conditions that might be related to the event. It's crucial for calculating posterior probabilities, which are the probabilities of hypotheses given observed evidence.

P ( A ∣ B ) is the posterior probability: the probability of hypothesis A given the evidence B .
P(B∣A) is the likelihood: the probability of observing evidence B given that hypothesis A is true.
P(A) is the prior probability: the initial probability of hypothesis A , before observing evidence B .
P(B) is the marginal probability: the total probability of observing evidence B B under all possible hypotheses.

Question 7: Describe how you would statistically determine if the results of an A/B test are significant - walk me through AB Testing process.

In this question, the interviewer is assessing your comprehensive knowledge of the A/B testing framework. They are looking for evidence that you can navigate the full spectrum of A/B testing procedures, which is essential for data scientists and AI professionals tasked with optimizing features, making data-informed decisions, and testing software products.

The interviewer wants to confirm that you understand each step in the process, beginning with formulating statistical hypotheses derived from business objectives. They are interested in your ability to conduct a power analysis and discuss its components, including determining effect size, significance level, and power, all critical in calculating the minimum sample size needed to detect a true effect and prevent p-hacking.

The discussion on randomization, data collection, and monitoring checks whether you grasp how to maintain the integrity of the test conditions. You should also be prepared to explain the selection of appropriate statistical tests, calculation of test statistics, p-values, and interpretation of results for both statistical and practical significance.

Ultimately, the interviewer is testing whether you can act as a data advocate: someone who can meticulously run A/B tests, interpret the results, and communicate findings and recommendations effectively to stakeholders, thereby driving data-driven decision-making within the organization.

To Learn AB Testing check my AB Testing Crash Course on YouTube .

In an A/B test, my first step is to establish clear business and statistical hypotheses. For example, if we’re testing a new webpage layout, the business hypothesis might be that the new layout increases user engagement. Statistically, this translates to expecting a higher mean engagement score for the new layout compared to the old.

Next, I’d conduct a power analysis. This involves deciding on an effect size that's practically significant for our business context—say, a 10% increase in engagement. I'd choose a significance level, commonly 0.05, and aim for a power of 80%, reducing the likelihood of Type II errors.

The power analysis, which takes into account the effect size, significance level, and power, helps determine the minimum sample size needed. This is crucial for ensuring that our test is adequately powered to detect the effect we care about and for avoiding p-hacking by committing to a sample size upfront.

With our sample size determined, I’d ensure proper randomization in assigning users to the control and test groups, to eliminate selection bias. During the test, I’d closely monitor data collection for any anomalies or necessary adjustments.

Upon completion of the data collection, I’d choose an appropriate statistical test based on the data distribution and variance homogeneity—typically a t-test if the sample size is small or a normal distribution can’t be assumed, or a Z-test for larger samples with a known variance.

Calculating the test statistic and the corresponding p-value allows us to test the null hypothesis. If the p-value is less than our chosen alpha level, we reject the null hypothesis, suggesting that the new layout has a statistically significant impact on engagement.

In addition to statistical significance, I’d evaluate the practical significance by looking at the confidence interval for the effect size and considering the business impact.

Finally, I’d document the entire process and results, then communicate them to stakeholders in a clear, non-technical language. This includes not just the statistical significance, but also how the results translate to business outcomes. As a data advocate, my goal is to support data-driven decisions that align with our business objectives and user experience strategy

For getting more interview questions from Stats to Deep Learning - with over 400 Q&A as well as personalized interview preparation check out our Free Resource Hub and our Data Science Bootcamp with Free Trial .

Thank you for choosing this guide as your learning companion. As you continue to explore the vast field of machine learning, I hope you do so with confidence, precision, and an innovative spirit. Best wishes in all your future endeavors!

About the Author

I am Tatev Aslanyan , Senior Machine Learning and AI Researcher, and Co-Founder of LunarTech where we are making Data Science and AI accessible to everyone. I have had the privilege of working in Data Science across numerous countries, including the US, UK, Canada, and the Netherlands.

With an MSc and BSc in Econometrics under my belt, my journey in Machine and AI has been nothing short of incredible. Drawing from my technical studies during my Bachelors & Masters, along with over 5 years of hands-on experience in the Data Science Industry, in Machine Learning and AI, I've gathered this high-level summary of ML topics to share with you.

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Hypothesis Testing
Hypothesis testing example. You want to test whether there is a relationship between gender and height. Based on your knowledge of human physiology, you formulate a hypothesis that men are, on average, taller than women. To test this hypothesis, you restate it as: H 0: Men are, on average, not taller than women. H a: Men are, on average, taller ...
S.3.3 Hypothesis Testing Examples
If the biologist set her significance level $\alpha$ at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):s-3-3. Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the null hypothesis.
4 Examples of Hypothesis Testing in Real Life
Example 1: Biology. Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals. For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than ...
7.1: Basics of Hypothesis Testing
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
Introduction to Hypothesis Testing
The null hypothesis, denoted as H 0, is the hypothesis that the sample data occurs purely from chance. The alternative hypothesis, denoted as H 1 or H a, is the hypothesis that the sample data is influenced by some non-random cause. Hypothesis Tests. A hypothesis test consists of five steps: 1. State the hypotheses. State the null and ...
Hypothesis Testing
One Sample Hypothesis Testing Examples: #3. Watch the video for an example of a two-tailed z-test: Blood glucose levels for obese patients have a mean of 100 with a standard deviation of 15. A researcher thinks that a diet high in raw cornstarch will have a positive or negative effect on blood glucose levels. A sample of 30 patients who have ...
Introduction to Hypothesis Testing with Examples
Likelihood ratio. In the likelihood ratio test, we reject the null hypothesis if the ratio is above a certain value i.e, reject the null hypothesis if L(X) > 𝜉, else accept it. 𝜉 is called the critical ratio.. So this is how we can draw a decision boundary: we separate the observations for which the likelihood ratio is greater than the critical ratio from the observations for which it ...
Significance tests (hypothesis testing)
Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.
S.3 Hypothesis Testing
S.3 Hypothesis Testing. In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail. The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data).
A Gentle Introduction to Statistical Hypothesis Testing
A statistical hypothesis test may return a value called p or the p-value. This is a quantity that we can use to interpret or quantify the result of the test and either reject or fail to reject the null hypothesis. This is done by comparing the p-value to a threshold value chosen beforehand called the significance level.
PDF Hypothesis Testing
23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis and the alternative hypothesis. 2. ... Collect and summarize data into a test statistic. Sample proportion is: 233/518 = 0.45. The standard deviation = (0.50) × (1 - 0.50) = 0.022. 518
Hypothesis Testing
Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant. It involves the setting up of a null hypothesis and an alternate hypothesis. There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
6a.2
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
9.E: Hypothesis Testing with One Sample (Exercises)
An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized-approximately 1,200 students-small city demographic) to determine if the local high school's percentage was lower. One hundred fifty students were chosen at random and surveyed.
What is Hypothesis Testing in Statistics? Types and Examples
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.
Simple hypothesis testing (practice)
Simple hypothesis testing. Niels has a Magic 8 -Ball, which is a toy used for fortune-telling or seeking advice. To consult the ball, you ask the ball a question and shake it. One of 5 different possible answers then appears at random in the ball. Niels sensed that the ball answers " Ask again later " too frequently.
Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example
Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used ...
Hypothesis testing and p-values (video)
In this video there was no critical value set for this experiment. In the last seconds of the video, Sal briefly mentions a p-value of 5% (0.05), which would have a critical of value of z = (+/-) 1.96. Since the experiment produced a z-score of 3, which is more extreme than 1.96, we reject the null hypothesis.
An Introduction to t Tests
Revised on June 22, 2023. A t test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another. t test example.
9.4 Full Hypothesis Test Examples
They believe that 50% of first-time borrowers take out smaller loans than other borrowers. They perform a hypothesis test to determine if the percentage is the same or different from 50%. They sample 100 first-time borrowers and find 53 of these loans are smaller that the other borrowers. For the hypothesis test, they choose a 5% level of ...
Hypothesis Examples: How to Write a Great Research Hypothesis
Examples of a complex hypothesis include: "People with high-sugar diets and sedentary activity levels are more likely to develop depression." "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."
Learn Statistics for Data Science, Machine Learning, and AI
The rejection region of this two-sided 2-sample Z-test can be visualized by the following graph: Image Source: The Author. The Python code snippet you've provided performs a two-sample Z-test for proportions. This type of test is used to determine whether there is a significant difference between the proportions of two groups.
How to Find P Value from a Test Statistic
Hypothesis tests are used to test the validity of a claim that is made about a population. This claim that's on trial, in essence, is called the null hypothesis (H 0). ... For example, when testing H o: p = 0.25 versus H a: p < 0.25, the p-value turns out to be 0.1056. This is because the test statistic was -1.25, and when you look this ...

Introduction to Hypothesis Testing

The Two Types of Statistical Hypotheses

Hypothesis Tests

The Two Types of Decision Errors

One-Tailed and Two-Tailed Tests

Types of Hypothesis Tests

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Hypothesis Testing

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9.E: Hypothesis Testing with One Sample (Exercises)

9.1: Introduction

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9.7: Hypothesis Testing of a Single Mean and Single Proportion

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What Is Hypothesis Testing in Statistics?

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Null Hypothesis and Alternate Hypothesis

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Step 2: Gather Data

Step 3: Conduct a Statistical Test

Step 4: Determine Rejection Of Your Null Hypothesis

Step 5: Present Your Results

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Simple and Composite Hypothesis Testing

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Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example