hypothesis test statistic formula

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.

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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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Test Statistics: Definition, Formulas & Examples

03.17.2022 • 10 min read

Sarah Thomas

Subject Matter Expert

This article explains what a test statistic is, how to complete one with formulas, and how to find the value for t-tests.

In This Article

What is a Standardized Test Statistic?

The general formula for calculating test statistics, types of test statistics with formulas, difference between t-tests and z-tests and when to use each, how to interpret a test statistic, don't overpay for college statistics.

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A test statistic is a standardized score used in hypothesis testing. It tells you how likely the results obtained from your sample data are under the assumption that the null hypothesis is true. The more unlikely your results are under this assumption, the easier it becomes to reject the null hypothesis in favor of an alternative hypothesis. The more likely your results are, the harder it becomes to reject the null hypothesis.

There are different kinds of test statistics, but they all work the same way. A test statistic maps the value of a particular sample statistic (such as a sample mean or a sample proportion) to a value on a standardized distribution, such as the Standard Normal Distribution or the t-distribution. This allows you to determine how likely or unlikely it is to observe the particular value of the statistic you obtained.

Olanrewaju Michael Akande reviews normal distribution in the following lesson clip:

As a quick example, say you have a null hypothesis that the average wait time to get seated at your favorite restaurant—at a table for two without a reservation on a Friday night—is 45 minutes. You select a random sample of 100 parties that got seated under these conditions and ask them what their wait times were. You find that the average wait time for your sample is 55 minutes ( x ˉ \bar{x} x ˉ = 55 minutes). A test statistic will convert this sample statistic x ˉ \bar{x} x ˉ into a standardized number that helps you answer this question:

“Assuming that my null hypothesis is true—assuming that the average wait time at the restaurant actually is 45 minutes—what is the likelihood that I found an average wait time of 55 minutes for my randomly drawn sample?”

Remember, the lower the likelihood of observing your sample statistic, the more confident you can be rejecting the null hypothesis.

The type of test statistic you use in a hypothesis test depends on several factors including:

The type of statistic you are using in the test

The size of your sample

Assumptions you can make about the distribution of your data

Assumptions you can make about the distribution of the statistic used in the test

The formula for calculating test statistics takes the following general form:

Remember, a statistic is a measure calculated from a single sample or many samples. Examples include the sample mean x ˉ \bar{x} x ˉ , the difference between two sample means x 1 ˉ − x 2 ˉ \bar{x_{1}} - \bar{x_{2}} x 1 ​ ˉ ​ − x 2 ​ ˉ ​ , or a sample proportion p ^ \hat{p} p ^ ​ .

A parameter is a measure calculated from a single population or many populations. Examples include the population mean μ \mu μ , the difference between two population means μ 1 − μ 2 \mu_{1}-\mu_{2} μ 1 ​ − μ 2 ​ , or a population proportion p p p .

In the denominator of the equation, you have the standard deviation—or the approximated standard deviation—of the statistic used in the numerator. If you use the sample mean x ˉ \bar{x} x ˉ , in the numerator, you should use the standard deviation of x ˉ \bar{x} x ˉ or an approximation of it in the denominator.

The test statistics you are most likely to encounter in an introductory statistics class are:

The Z-test statistic for a single sample mean

The Z-test statistic for population proportions

The t-test statistic for a single sample mean

The t-test statistic for two sample means

Z-test for a Sample Mean

We use the Z-test statistic (or Z-statistic) for a sample mean in hypothesis tests involving a sample mean x ˉ \bar{x} x ˉ , calculated for a single sample.

You use this test statistic when:

Your sample size is greater than or equal to 30 (n ≥ \geq ≥ 30)

The sampling distribution of the sample mean is assumed to be normal

The standard deviation of the population parameter σ \sigma σ is known.

The formula for this type of Z-test statistic is:

Z Z Z is the symbol for the Z-test statistic

x ˉ \bar{x} x ˉ is the sample mean

μ 0 \mu_{0} μ 0 ​ is the hypothesized value of the population mean according to the null hypothesis

σ \sigma σ is the population standard deviation

n n n is the sample size

σ n \frac{\sigma}{\sqrt{n}} n ​ σ ​ is the standard error of x ˉ \bar{x} x ˉ . The standard error is just the standard deviation of the sampling distribution of the sample mean.

You may notice that a Z-test statistic is just a z-score for a particular value of a normally distributed statistic. There are many variations of the Z-test statistic. We can use these in hypothesis tests, where the sample statistic is being used in the test is approximately normally distributed. One such variation of the Z-test statistic is the Z-test for proportions.

Z-test for Proportions

We use the Z-test statistic for proportions in hypothesis tests where a sample proportion p ^ \hat{p} p ^ ​ is being tested against the hypothesized value of the population proportion, p 0 p_{0} p 0 ​ . We use the Z-test for proportions when your sample size is greater than or equal to 30 (n ≥ \geq ≥ 30), and the distribution of the sample statistic is assumed to be normal. The formula for the Z-test statistic for population proportions is:

Z is the symbol for the Z-test statistic for population proportions

p ^ \hat{p} p ^ ​ is the sample proportion

p 0 p_{0} p 0 ​ is the hypothesized value of the population proportion according to the null hypothesis

When your sample size is smaller than 30 (n<30)—or when you cannot assume that the distribution of your sample statistic is normally distributed—you’ll often use a t-test statistic rather than a Z-test.

T-test for a Single Sample Mean

We use the t-test statistic (or t-statistic) for a sample mean in hypothesis tests involving a sample mean calculated for a single sample drawn from a population. Unlike the Z-test for a single sample mean, you use the t-test when:

Your sample size is less than 30 (n<30)

The distribution of the sample statistic is not approximated by a normal distribution

The standard deviation of the population parameter σ \sigma σ is unknown

A t-test statistic maps your statistics to a t-distribution as opposed to the normal distribution with a Z-test. A t-distribution is like a standard normal distribution, but it has thicker tails and changes depending on your sample size n n n . When n n n is large, the t-distribution is closer to the normal distribution; and as the sample size gets larger and larger, a t-distribution will converge to the normal distribution. As n n n gets smaller, the t-distribution gets flatter with thicker tails.

The formula for the t-test statistic for a sample mean is:

t t t is the symbol for the t-test statistic

μ 0 \mu_0 μ 0 ​ is the value of the population mean according to the null hypothesis

s s s is the sample standard deviation

s n \frac{s}{\sqrt{n}} n ​ s ​ is an approximation of the standard error of x ˉ \bar{x} x ˉ . In a t-test, because you do not know the value of the population standard deviation, you need to approximate the standard error of x ˉ \bar{x} x ˉ using the sample standard deviation s s s .

T-test for Two Sample Means

We can also use t-test statistics in hypothesis tests where the values of two sample means ( x 1 x_{1} x 1 ​ and x 2 x_{2} x 2 ​ ) are being compared. You do this to test the null hypothesis that the two samples are drawn from the same underlying population. If the null hypothesis is true, then any difference between the sample means is due to random variations in the data. Rejecting the null hypothesis suggests that the samples were drawn from two distinct populations and that the difference in the sample means reflects actual differences in the characteristics of subjects in one population compared to the other.

Like the t-test for a single sample mean, you use the t-test for two sample means when:

Your sample sizes are less than 30 (n<30)

The distribution of the sample statistics are not approximated by a normal distribution

The formula for the t-test statistic for two sample means is‌:

x 1 ˉ \bar{x_1} x 1 ​ ˉ ​ is the sample mean of sample 1

x 2 ˉ \bar{x_2} x 2 ​ ˉ ​ is the sample mean of sample 2

μ 1 \mu_1 μ 1 ​ is the mean of the population from which sample 1 was drawn

μ 2 \mu_2 μ 2 ​ is the mean of the population from which sample 2 was drawn

s 1 2 s_1^2 s 1 2 ​ is the variance of sample 1

s 2 2 s_2^2 s 2 2 ​ is the variance of sample 2

n 1 n_{1} n 1 ​ is the sample size for sample 1

n 2 n_{2} n 2 ​ is the sample size for sample 2

T-tests are generally used in place of Z-tests when one or more of the following conditions hold: The sample size is less than 30 (n \sigma is unknown

If you know the population standard deviation σ \sigma σ and you are confident that the statistic used in your hypothesis test is normally distributed, then you can use a Z-test.

As with all test statistics, you should only use a Z-test or a t-test when your data is from a randomly and independently drawn sample.

We use test statistics together with critical values, p-values, and significance levels to determine whether to reject or not a null hypothesis.

A critical value is a value of a test statistic that marks a cutoff point. If a test statistic is more extreme than the critical value—greater than the critical value in the right tail of a distribution or less than the critical value in the left tail of a distribution—the null hypothesis is rejected.

Critical values are determined by the significance level (or alpha level) of a hypothesis test. The significance level you use is up to you, but the most commonly used significance level is 0.05 ( α \alpha α =0.05).

A significance level of 0.05 means that if the probability of observing a sample statistic at least as extreme as the one you observed is less than 0.05 (or 5%), you should reject your null hypothesis. In a one-sided hypothesis test that uses a Z-test statistic, a significance level of 0.05 is associated with a critical value of 1.645 when you conduct the test in the right tail and a value of -1.645 when you conduct the test in the left tail.

A p-value is the probability associated with your test statistic’s value. Let’s say you calculate a Z-test statistic that maps to the standard normal distribution. You find that the test statistic is equal to 1.75. For this ‌value of a Z-test statistic, the associated p-value is 0.04 or 4%—you can find p-values using tables or statistical software.

A p-value of 0.04 means that the probability of observing a sample statistic at least as extreme as the one you found from your sample data is 4%. If you choose a significance level of 0.05 for your test, we would reject the null hypothesis, since the p-value of 0.04 is less than the significance level of 0.05.

It can be easy to confuse test statistics, critical values, significance levels, and p-values. Remember, these are all different measures involved in determining whether to reject or fail to reject a null hypothesis.

Critical values and significance levels provide cut-offs for your test. The difference between a critical value and a significance level is that the critical value is a point on the distribution, and the significance level is a probability represented by an area under the distribution.

You can compare the test statistic and the p-value against the critical value and the significance level.

If the test statistic is more extreme than the critical value, you reject the null hypothesis.

If the p-value is less than the significance level, you reject the null hypothesis.

If the test statistic is less extreme than the critical value, you fail to reject the null hypothesis.

If the p-value is greater than the significance level, you reject the null hypothesis.

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Unit 12: Significance tests (hypothesis testing)

About this unit.

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.

The idea of significance tests

• Simple hypothesis testing (Opens a modal)
• Idea behind hypothesis testing (Opens a modal)
• Examples of null and alternative hypotheses (Opens a modal)
• P-values and significance tests (Opens a modal)
• Comparing P-values to different significance levels (Opens a modal)
• Estimating a P-value from a simulation (Opens a modal)
• Using P-values to make conclusions (Opens a modal)
• Simple hypothesis testing Get 3 of 4 questions to level up!
• Writing null and alternative hypotheses Get 3 of 4 questions to level up!
• Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

• Introduction to Type I and Type II errors (Opens a modal)
• Type 1 errors (Opens a modal)
• Examples identifying Type I and Type II errors (Opens a modal)
• Introduction to power in significance tests (Opens a modal)
• Examples thinking about power in significance tests (Opens a modal)
• Consequences of errors and significance (Opens a modal)
• Type I vs Type II error Get 3 of 4 questions to level up!
• Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

• Constructing hypotheses for a significance test about a proportion (Opens a modal)
• Conditions for a z test about a proportion (Opens a modal)
• Reference: Conditions for inference on a proportion (Opens a modal)
• Calculating a z statistic in a test about a proportion (Opens a modal)
• Calculating a P-value given a z statistic (Opens a modal)
• Making conclusions in a test about a proportion (Opens a modal)
• Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
• Conditions for a z test about a proportion Get 3 of 4 questions to level up!
• Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
• Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
• Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

• Writing hypotheses for a significance test about a mean (Opens a modal)
• Conditions for a t test about a mean (Opens a modal)
• Reference: Conditions for inference on a mean (Opens a modal)
• When to use z or t statistics in significance tests (Opens a modal)
• Example calculating t statistic for a test about a mean (Opens a modal)
• Using TI calculator for P-value from t statistic (Opens a modal)
• Using a table to estimate P-value from t statistic (Opens a modal)
• Comparing P-value from t statistic to significance level (Opens a modal)
• Free response example: Significance test for a mean (Opens a modal)
• Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
• Conditions for a t test about a mean Get 3 of 4 questions to level up!
• Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
• Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
• Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

More significance testing videos

• Hypothesis testing and p-values (Opens a modal)
• One-tailed and two-tailed tests (Opens a modal)
• Z-statistics vs. T-statistics (Opens a modal)
• Small sample hypothesis test (Opens a modal)
• Large sample proportion hypothesis testing (Opens a modal)

Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

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Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

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8.2: Hypothesis Testing of Single Proportion

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Learning Objectives

• To learn how to apply the five-step critical value test procedure for test of hypotheses concerning a population proportion.
• To learn how to apply the five-step \(p\)-value test procedure for test of hypotheses concerning a population proportion.

Both the critical value approach and the p-value approach can be applied to test hypotheses about a population proportion p. The null hypothesis will have the form \(H_0 : p = p_0\) for some specific number \(p_0\) between \(0\) and \(1\). The alternative hypothesis will be one of the three inequalities

• \(p <p_0\),
• \(p>p_0\), or
• \(p≠p_0\)

for the same number \(p_0\) that appears in the null hypothesis.

The information in Section 6.3 gives the following formula for the test statistic and its distribution. In the formula \(p_0\) is the numerical value of \(p\) that appears in the two hypotheses, \(q_0=1−p_0, \hat{p}\) is the sample proportion, and \(n\) is the sample size. Remember that the condition that the sample be large is not that \(n\) be at least 30 but that the interval

\[ \left[ \hat{p} −3 \sqrt{ \dfrac{\hat{p} (1−\hat{p} )}{n}} , \hat{p} + 3 \sqrt{ \dfrac{\hat{p} (1−\hat{p} )}{n}} \right] \nonumber \]

lie wholly within the interval \([0,1]\).

Standardized Test Statistic for Large Sample Hypothesis Tests Concerning a Single Population Proportion

\[ Z = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0q_o}{n}}} \label{eq2} \]

The test statistic has the standard normal distribution.

The distribution of the standardized test statistic and the corresponding rejection region for each form of the alternative hypothesis (left-tailed, right-tailed, or two-tailed), is shown in Figure \(\PageIndex{1}\).

Example \(\PageIndex{1}\)

A soft drink maker claims that a majority of adults prefer its leading beverage over that of its main competitor’s. To test this claim \(500\) randomly selected people were given the two beverages in random order to taste. Among them, \(270\) preferred the soft drink maker’s brand, \(211\) preferred the competitor’s brand, and \(19\) could not make up their minds. Determine whether there is sufficient evidence, at the \(5\%\) level of significance, to support the soft drink maker’s claim against the default that the population is evenly split in its preference.

We will use the critical value approach to perform the test. The same test will be performed using the \(p\)-value approach in Example \(\PageIndex{3}\).

We must check that the sample is sufficiently large to validly perform the test. Since \(\hat{p} =270/500=0.54\),

\[\begin{align} & \left[ \hat{p} −3\sqrt{ \dfrac{\hat{p} (1−\hat{p} )}{n}} ,\hat{p} +3\sqrt{ \dfrac{\hat{p} (1−\hat{p} )}{n}} \right] \\ &=[0.54−(3)(0.02),0.54+(3)(0.02)] \\ &=[0.48, 0.60] ⊂[0,1] \end{align} \nonumber \]

so the sample is sufficiently large.

• Step 1. The relevant test is

\[H_0 : p = 0.50  \nonumber \]

\[vs. \nonumber \]

\[H_a : p > 0.50\, @ \,\alpha =0.05 \nonumber \]

where \(p\) denotes the proportion of all adults who prefer the company’s beverage over that of its competitor’s beverage.

• Step 2. The test statistic (Equation \ref{eq2}) is

\[Z=\dfrac{\hat{p} −p_0}{\sqrt{ \dfrac{p_0q_0}{n}}} \nonumber \]

and has the standard normal distribution.

• Step 3. The value of the test statistic is

\[ \begin{align} Z &=\dfrac{\hat{p} −p_0}{\sqrt{ \dfrac{p_0q_0}{n}}} \\[6pt] &= \dfrac{0.54−0.50}{\sqrt{\dfrac{(0.50)(0.50)}{500}}} \\[6pt] &=1.789 \end{align} \nonumber \]

• Step 4. Since the symbol in \(H_a\) is “\(>\)” this is a right-tailed test, so there is a single critical value, \(z_{α}=z_{0.05}\). Reading from the last line in Figure 7.1.6 its value is \(1.645\). The rejection region is \([1.645,∞)\).
• Step 5. As shown in Figure \(\PageIndex{2}\) the test statistic falls in the rejection region. The decision is to reject \(H_0\). In the context of the problem our conclusion is:

The data provide sufficient evidence, at the \(5\%\) level of significance, to conclude that a majority of adults prefer the company’s beverage to that of their competitor’s.

Example \(\PageIndex{2}\)

Globally the long-term proportion of newborns who are male is \(51.46\%\). A researcher believes that the proportion of boys at birth changes under severe economic conditions. To test this belief randomly selected birth records of \(5,000\) babies born during a period of economic recession were examined. It was found in the sample that \(52.55\%\) of the newborns were boys. Determine whether there is sufficient evidence, at the \(10\%\) level of significance, to support the researcher’s belief.

We will use the critical value approach to perform the test. The same test will be performed using the \(p\)-value approach in Example \(\PageIndex{1}\).

The sample is sufficiently large to validly perform the test since

\[\sqrt{ \dfrac{\hat{p} (1−\hat{p} )}{n}} =\sqrt{ \dfrac{(0.5255)(0.4745)}{5000}} ≈0.01 \nonumber \]

\[\begin{align} & \left[ \hat{p} −3\sqrt{ \dfrac{\hat{p} (1−\hat{p} )}{n}} ,\hat{p} +3\sqrt{ \dfrac{\hat{p} (1−\hat{p} )}{n}} \right] \\ &=[0.5255−0.03,0.5255+0.03] \\ &=[0.4955,0.5555] ⊂[0,1] \end{align} \nonumber \]

• Step 1 . Let \(p\) be the true proportion of boys among all newborns during the recession period. The burden of proof is to show that severe economic conditions change it from the historic long-term value of \(0.5146\) rather than to show that it stays the same, so the hypothesis test is

\[H_0 : p = 0.5146  \nonumber \]

\[H_a : p \neq 0.5146\, @ \,\alpha =0.10 \nonumber \]

\[ \begin{align} Z &=\dfrac{\hat{p} −p_0}{\sqrt{ \dfrac{p_0q_0}{n}}} \\[6pt] &= \dfrac{0.5255−0.5146}{\sqrt{\dfrac{(0.5146)(0.4854)}{5000}}} \\[6pt] &=1.542 \end{align} \nonumber \]

• Step 4. Since the symbol in \(H_a\) is “\(\neq\)” this is a two-tailed test, so there are a pair of critical values, \(\pm z_{\alpha /2}=\pm z_{0.05}=\pm 1.645\). The rejection region is \((-\infty ,-1.645]\cup [1.645,\infty )\).
• Step 5. As shown in Figure \(\PageIndex{3}\) the test statistic does not fall in the rejection region. The decision is not to reject \(H_0\). In the context of the problem our conclusion is:

The data do not provide sufficient evidence, at the \(10\%\) level of significance, to conclude that the proportion of newborns who are male differs from the historic proportion in times of economic recession.

Example \(\PageIndex{3}\)

Perform the test of Example \(\PageIndex{1}\) using the \(p\)-value approach.

We already know that the sample size is sufficiently large to validly perform the test.

• Steps 1–3 of the five-step procedure described in Section 8.3 have already been done in Example \(\PageIndex{1}\) so we will not repeat them here, but only say that we know that the test is right-tailed and that value of the test statistic is \(Z = 1.789\).
• Step 4. Since the test is right-tailed the p-value is the area under the standard normal curve cut off by the observed test statistic, \(Z = 1.789\), as illustrated in Figure \(\PageIndex{4}\). By Figure 7.1.5 that area and therefore the p-value is \(1−0.9633=0.0367\).
• Step 5. Since the \(p\)-value is less than \(α=0.05\) the decision is to reject \(H_0\).

Example \(\PageIndex{4}\)

Perform the test of Example \(\PageIndex{2}\) using the \(p\)-value approach.

• Steps 1–3 of the five-step procedure described in Section 8.3 have already been done in Example \(\PageIndex{2}\). They tell us that the test is two-tailed and that value of the test statistic is \(Z = 1.542\).
• Step 4. Since the test is two-tailed the \(p\)-value is the double of the area under the standard normal curve cut off by the observed test statistic, \(Z = 1.542\). By Figure 7.1.5 that area is \(1-0.9382=0.0618\), as illustrated in Figure \(\PageIndex{5}\), hence the \(p\)-value is \(2\times 0.0618=0.1236\).
• Step 5. Since the \(p\)-value is greater than \(\alpha =0.10\) the decision is not to reject \(H_0\).

Key Takeaway

• There is one formula for the test statistic in testing hypotheses about a population proportion. The test statistic follows the standard normal distribution.
• Either five-step procedure, critical value or \(p\)-value approach, can be used.

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T-Test in Statistics: Formula, Types and Steps

• Data Types in Statistics
• What are the types of statistics?
• Introduction of Statistics and its Types
• What are some of the important formulae used in statistics?
• Type I and Type II Errors
• Statistics Formulas
• How to Calculate the P-Value of an F-Statistic in R
• How to Conduct a Paired Samples T-Test in Python
• Types of Statistical Series
• Mode in Statistics | Definition, Formula and Examples
• Upper Tail Test of Population Proportion in R
• Differences Between two-sample, t-test and paired t-test
• Measures of Central Tendency in Statistics
• Who is the Father of Statistics?
• What is n in Statistics?
• How to Conduct a Two Sample T-Test in Python
• How to Conduct a One Sample T-Test in Python
• SciPy - Statistical Significance Tests
• How to Conduct a Sobel Test in R
• How to Perform an F-Test in Python
• Hypothesis Testing Formula
• Two-Tailed Test of Population Proportion in R
• F-Test in Statistics
• Student's t-distribution in Statistics
• Python - Student’s t Distribution in Statistics
• TCS SDE Sheet: Interview Questions and Answers
• sympy.stats.StudentT() in Python
• Tukey-Kramer Test for Post Hoc Analysis
• Decision Table Based Testing in Software Testing

T-Test is a method used in statistics to determine if there is a significant difference between the means of two groups and how they are related. In T-Test statistics, the sample data is a subset of the two groups that we use to draw conclusions about the groups as a whole. 

T-Test in Statistics

For example, if we want to know the average weight of mangoes grown on a farm, the population would consist of all the mangoes that grew on the farm. However, it would be time-consuming to weigh each mango. Instead, we could take a sample of mangoes from trees at different locations on the farm and use their weights to make inferences about the average weight of all the mangoes grown on the farm.

T-Test Definition

T-Test is a statistics method to determine significance changes between means of two groups. It helps us to determine whether the data sets belong to the same group or not. This comparison is often called a T-test.

T-Test Formula

There is no specific formula for T-Test, as it is divided into various types such as One Samples T-Test, Independent Samples T-test, etc. which are used as per the need. The formula used in each type is defined under the specific headings. The formula allows us to calculate a T-value which helps to make a comparison between the data sets.

Value of T-Test

The value obtained by substituting required values to the t-test formula is called the t-value. A larger T-value implies that the sets belong to a different population, while a smaller T-value implies that they belong to the same population. The formula is comprised of the values of mean, standard deviation and variance of the data sets under consideration.

How to Calculate T Value in T-Test

To calculate T-value in T-Test, we can use the following steps:

Step 1: To perform a T-test, two hypotheses namely the null hypothesis and the alternative hypothesis are defined which have different meanings for different types of T-tests.

Step 2: And, a value for the level of significance is defined which signifies the probability of making a Type I error, which implies the rejection of the null hypothesis while it is actually true. Commonly used values of level of significance are 0.05 (5%) and 0.01 (1%).

Step 3: A higher significance level, such as α = 0.05, provides a higher tolerance for Type I errors, meaning that it is more likely to reject the null hypothesis even when it is true.

Step 4: On the other hand, a lower significance level, such as α = 0.01, reduces the risk of Type I errors but it may increase the chances of accepting the null hypothesis when it is actually false, resulting in a Type II error.

Types of T-Test

Below are the three types of T-Test mentioned below.

• One Sample T-test
• Independent Samples T-test
• Paired Samples T-test

Let’s discuss these types in detail as follows:

One Sample T-Test

As the name implies, this test is used when we have one data set for a sample and we need to determine whether this data set belongs to a particular population or not. The mean value for the population data must be known in this case. The formula to determine T-value, in this case, is as follows:

t = (x̄ – μ) / (σ / √n) Where, t is the t-value, x̄ is the Sample mean, μ is the Population mean, σ is the Sample standard deviation, and n is the Sample size.

Steps to Calculate T Value One Sample T-Test 

To perform the One Sample T-test, the steps listed below are generally followed:

Step 1: State a null hypothesis and an alternative hypothesis. The null hypothesis assumes that the sample mean and the known population mean (μ) are equal, while the other assumes that the sample mean is different from the population mean.

Step 2: Define values for the level of significance (α) and the degree of freedom (df). The degree of freedom equals (n – 1) for this case.

Step 3: Calculate the t-value using the formula stated above by putting all the known values of the sample mean (x̄), sample standard deviation (σ), the population mean (μ), and the sample size (n).

Step 4: Determine the associated p-value with the t-value using a t-distribution table.

Step 5: Compare the p-value to the level of significance. If the p-value is less than the level of significance, reject the null hypothesis and conclude that the sample mean is significantly different from the population mean. Otherwise, conclude that there is no significant difference between the sample mean and the population mean.

Independent Samples T-Test

As the name suggests, an Independent samples T-test is used when we need to compare the statistical means of two independent samples or groups. It helps us determine whether there is a significant difference between the means of the two groups. If there is a significant difference, it suggests that the groups likely have different population means; otherwise, they have the same population means.

For example, when an investigation aims to determine if there is a significant difference in the mean scores between athletes who follow a specific training camp (Team A) and those who do not (Team B), an independent samples t-test can be conducted.

This test is performed using either of two assumptions made about variances of the samples, one assumes equal variances for the sample and the other assumes unequal variances for the samples.

Unequal Variances T-Test

Under this test, variances of two groups considered are assumed to be equal. This is appropriate when we are uncertain about the variances of the two groups considered. The formula to calculate T-value, in this case, is as follows:

t = (x̄ 1 – x̄ 2 ) / √((σ 1 2 /n 1 ) + (σ 2 2 /n 2 )) Where, x̄ 1 is the sample mean of Group 1, x̄ 2 is the sample mean of Group 2, σ 1 is the sample standard deviation of Group 1, σ 2 is the sample standard deviation of Group 2, n 1 is the sample size of Group 1, and n 2 sample size of Group 2.

Equal Variance T-Test

Under this test, variances of two groups considered are assumed to be equal. This is appropriate when we have some assurance about variances of data considered to be equal. The formula to calculate T-value, in this case, is similar to the above formula with a slight change that σ 1 = σ 2 = σ.

t = (x̄ 1 – x̄ 2 ) / √(σ 2 (1/n 1 + 1/n 2 )) Where, x̄ 1 is the sample mean of Group 1, x̄ 2 is the sample mean of Group 2, σ is the standard deviation of both groups, n 1 is the sample size of Group 1, and n 2 sample size of Group 2.

T Test for Independent Samples

The steps listed below are generally followed to perform this test:

Step 1: State a null hypothesis and an alternate hypothesis. The null hypothesis assumes that the means of the two groups are equal (x̄ 1 = x̄ 2 ), while the other assumes that the means of the two groups are significantly different (x̄ 1 ≠ x̄ 2 ).

Step 2: Define the values for the level of significance (α) and the degrees of freedom (df). The degree of freedom equals (n 1 + n 2 – 2) in this case.

Step 3: Calculate the t-value from the formula defined above after obtaining the required data related to each group.

Step 4: Find the critical t-value from a t-distribution table with the corresponding degrees of freedom and level of significance.

Step 5: If the calculated t-value is greater than the critical t-value, then reject the null hypothesis. This indicates that there is a significant difference between the means of the two groups. Otherwise, the null hypothesis is not rejected. And, this suggests that there is no significant difference between the means of the two groups.

Paired Samples T-Test

The Paired samples t-test is used when we want to compare the means of two related groups or samples. For example, we may use this test to compare the average scores of the players of an athletics team before and after a training program. To calculate the t-value in this case, the following formula is used,

t = (x̄ d – μ d ) / (σ d / √n) Where: t is the t-value, x̄ d is the sample mean of the differences between the paired observations, μ d isthe population mean difference, σ d is the sample standard deviation of the differences, n is the number of paired observations.

Steps for Paired Samples T-Test

Following are the steps to perform this type of T-test:

Step 1: State the null hypothesis which assumes that there is no significant difference between the statistical means of the paired observations (μ d = 0) while the alternative hypothesis assumes that there is a significant difference between the statistical means of the paired observations (μ d ≠ 0).

Step 2: Match each observation in one group with a corresponding observation in the other group.

Step 3: Calculate the differences between each paired observation and then, calculate the mean of the differences (x̄ d ), and the sample standard deviation of the differences (σ d ). Furthermore, calculate the t-value from the formula.

Step 4: Obtain the critical t-value from a t-distribution table corresponding to the chosen level of significance (α) and degree of freedom (df). The degree of freedom (df) equals (n – 1) in this case.

Step 5: If the calculated t-value is greater than the critical t-value, then reject the null hypothesis. This indicates a significant difference in the sample before and after the intervention. Otherwise, it can be concluded that there is no significant difference in the sample before and after the intervention.

T-Distribution Table

A T-Distribution table is used to obtain a critical t-value that is used as a reference to the calculated t-value for obtaining further results. Critical t-value depends on values of the level of significance and degrees of freedom. A concise form of the table for critical t-values is as follows for your reference:

Solved Problems of T-Test Formula

Problem 1: Determine whether the average weight of a sample of 20 mangoes is significantly different from the population’s average weight of 70 grams. The sample mean weight is 70.55 grams, and the sample standard deviation is 2.82 grams. Use one sample T-test.

To perform a T-test, first of all, we define two hypotheses: Null hypothesis: The sample mean weight of mangoes is equal to the known population mean. (i.e., 70 grams). Alternative hypothesis: The sample mean weight of mangoes is not equal to the known mean value. Then, determine the degrees of freedom (d f ): d f = n – 1 = 20 – 1 = 19 and define the level of significance(α) as 0.05 for this case. Next, determine the t-value from the formula, t = (70.55 – 70) / (2.82 / √20) ⇒ t ≈ 1.041 From the t-distribution table, we find 1.041 < 2.093. (i.e. p-value for α = 0.05). So, the null hypothesis is true. Thus, we conclude that the sample does not vary significantly from the population.

Problem 2: Determine if there is a significant difference in the average scores between the two teams. The following data is given:  Team A: Score: 65, 68, 70, 63, 67 Team B: Score: 62, 66, 69, 64, 68

According to the question, we come to know that we need to perform an Independent Samples T-test. Set up the null hypothesis and alternative hypothesis: Null hypothesis: The means of the two groups are equal (μ A = μ B ). Alternative hypothesis: The means of the two groups are not equal (μ A ≠ μ B ). Next, we calculate the sample means (x̄ A and x̄ B ) and sample standard deviations (σ A and σ B ): Team A: Sample size (n A ) = 5 Sample mean (x̄ A ) = (65 + 68 + 70 + 63 + 67) / 5 = 66.6 Sample standard deviation (σ A ) ≈ 2.607 Team B: Sample size (n B ) = 5 Sample mean (x̄ B ) = (62 + 66 + 69 + 64 + 68) / 5 = 65.8 Sample standard deviation (σ B ) ≈ 2.588 Now, we calculate the t-value using the formula: t = (x̄ A – x̄ B ) / √((σ A 2 / n A ) + (σ B 2 / n B )) ⇒ t = (66.6 – 65.8) / √{(2.607 2 /5) + (2.588 2 /5)} ⇒ t ≈ 0.296 Then, determine the degrees of freedom (df):  df = n A + n B – 2 = 5 + 5 – 2 = 8  and set the level of significance as 0.05.  From the table, we get the critical t-value as 2.306. As the calculated t-value is less than the critical t-value, we conclude that the null hypothesis is not rejected, which suggests that there is no significant difference between the average scores of the two teams.

Problem 3: You need to assess the effectiveness of a new teaching scheme by comparing the test scores of the same group of students before and after the implementation of the scheme. The following data is given: Before scores: 76, 88, 65, 56, 76 After scores: 85, 95, 75, 60, 81 Determine if there is a significant difference in the average test scores before and after the implementation of the scheme.

Here, we need to perform a Paired Samples T-test, as we need to compare data of the same sample. Set up the null hypothesis and alternative hypothesis: Null hypothesis: The population mean difference between the before and after scores are zero (μ d = 0). Alternative hypothesis: The population mean difference between the before and after scores is not zero (μ d ≠ 0). Next, calculate the differences between the paired observations: Difference (d) = After score – Before score d 1 = 85 – 76 = 9 d 2 = 95 – 88 = 7 d 3 = 75 – 65 = 10 d 4 = 60 – 56 = 4 d 5 = 81 – 76 = 5 Now, calculate the sample mean (x̄d) and sample standard deviation (σ d ) of the differences: Sample size (n) = 5 Sample mean (x̄ d ) = (d 1 + d 2 + d 3 + d 4 + d 5 ) / 5 = (9 + 7 + 10 + 4 + 5) / 5 = 7 Sample standard deviation (σ d ) ≈ 2.828 Then, calculate the t-value using the formula: t = (x̄ d – μ d ) / (σ d / √n)  ⇒ t = (7 – 0) / (2.828 / √5) ⇒ t ≈ 5.535 Next, calculate the value of degrees of freedom (df):  df = n – 1 = 5 – 1 = 4.  And, define the level of significance(α) as 0.05.  Now, from the t-distribution table, we find that the critical t-value is 2.776. As the calculated t-value is greater than the critical t-value ( 5.535 > 2.776), thus, the null hypothesis is rejected. And we conclude that there is a significant difference in the average test scores before and after the implementation of the scheme.

T-Test in Statistics – FAQs

What is a t-test in statistics.

T-Test is the test in statistics to derive some conclusions for a population which is based upon some sample data using values of means and variances. 

When is a T-Test used?

The test is basically used to determine whether there is any significant difference in the statistical means of two samples of the data considered. The purpose to determine this can be to check if a sample data set belongs to the population data set, or if there is an effect of any variation on the data values before or after any specific treatment/intervention.

What are the Different Types of T-Tests?

There are three types of T-tests that are used as per the situation, listed as follows: One-sample T-test: It is used when we need to compare the mean of a single sample to a known (or assumed) population mean value. Independent T-test: It is used when we need to compare the means of two independent groups. Paired T-test: It is used to compare the means of two related or paired groups.

What does the T-Value obtain from the T-Test Formula Indicate?

The t-value indicates the magnitude of the difference between the means relative to the variability within the groups. A larger t-value suggests a greater difference between the means.

Are there any Assumptions related to Sample Data in Performing a T-Test on it?

The t-test assumes that the data within each group are normally distributed, the variances of the two groups are equal (in the case of an independent t-test), the observations are independent, and the data points represent their respective populations.

What are the Limitations of the T-Test?

The t-test assumes that the data meet the assumptions of normality, independence, and equal variances (in the case of an independent t-test). If these assumptions are not true, it can lead to inaccurate or misleading results. Also, the test is sensitive to outliers, and may not give accurate results for small sample sizes.

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S.3.2 hypothesis testing (p-value approach).

The P -value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis was true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P -value is small, say less than (or equal to) \(\alpha\), then it is "unlikely." And, if the P -value is large, say more than \(\alpha\), then it is "likely."

If the P -value is less than (or equal to) \(\alpha\), then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the P -value is greater than \(\alpha\), then the null hypothesis is not rejected.

Specifically, the four steps involved in using the P -value approach to conducting any hypothesis test are:

• Specify the null and alternative hypotheses.
• Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
• Using the known distribution of the test statistic, calculate the P -value : "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?")
• Set the significance level, \(\alpha\), the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the P -value to \(\alpha\). If the P -value is less than (or equal to) \(\alpha\), reject the null hypothesis in favor of the alternative hypothesis. If the P -value is greater than \(\alpha\), do not reject the null hypothesis.

Example S.3.2.1

Mean gpa section  .

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * equaling 2.5. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

Right Tailed

The P -value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the probability that we would observe a test statistic greater than t * = 2.5 if the population mean \(\mu\) really were 3. Recall that probability equals the area under the probability curve. The P -value is therefore the area under a t n - 1 = t 14 curve and to the right of the test statistic t * = 2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than \(\alpha\) = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ > 3 if we lowered our willingness to make a Type I error to \(\alpha\) = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

Left Tailed

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the probability that we would observe a test statistic less than t * = -2.5 if the population mean μ really were 3. The P -value is therefore the area under a t n - 1 = t 14 curve and to the left of the test statistic t* = -2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ < 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

In our example concerning the mean grade point average, suppose again that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean μ really was 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (hence the name "two-tailed" test). The P -value is, therefore, the area under a t n - 1 = t 14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually.

Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests. The P -value, 0.0254, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0254, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ ≠ 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0254, is then greater than \(\alpha\) = 0.01.

Now that we have reviewed the critical value and P -value approach procedures for each of the three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.

The good news is that, whenever possible, we will take advantage of the test statistics and P -values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.