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Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

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.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10, idea behind hypothesis testing.

• Examples of null and alternative hypotheses
• Writing null and alternative hypotheses
• P-values and significance tests
• Comparing P-values to different significance levels
• Estimating a P-value from a simulation
• Estimating P-values from simulations
• Using P-values to make conclusions

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Video transcript

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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9.6 Skills Focus: Selecting an Appropriate Inference Procedure

5 min read • january 8, 2023

Jed Quiaoit

One of the most important skills in AP Statistics is being able to identify the best inference procedure to use in order to complete a hypothesis test or confidence interval. We have covered all of the following types of procedures: 📄

One Proportion Z Test

One Proportion Z Interval

One Sample T Test

One Sample T Interval

Matched Pairs T Test

Two Proportion Z Test

Two Proportion Z Interval

Two Sample T Test

Two Sample T Interval

Chi Squared Goodness of Fit Test

Chi Squared Test for Independence

Chi Squared Test for Homogeneity

Linear Regression T Interval

Linear Regression T Test

For example, If given a problem involving one of the linear regression t procedures, it is most common that you will be given a computer output and be asked to make a conclusion or construct an interval.

Here are a couple illustrative flowchart "cheat sheets" on picking the right inferential procedure. Good luck! ⭐

Source: Mr. Sardinha

Source: Reddit

Here is a computer output similar to what you would see on the AP test. This is based on a study with a sample size of 30.

Remember from Unit 2, that we are only focusing on the inference values associated with the slope, which is the row entitled “Sick Days.”

Confidence Interval

In order to construct a confidence interval like we discussed in Section 9.2, we will need the point estimate (sample slope), t-score and standard error.

Everything except our t-score is given in the computer output, so we have to calculate our t-score based on our confidence level and sample size. We will first calculate our degrees of freedom of 28 and then use that with the invT function to calculate our t-score . We get a t-score of 2.05 for a 95% confidence level.

For the computer output above, our confidence interval would be:

0.962.05(0.12)

Which comes out to be (0.714, 1.206).

In this case, we can be sure that the two variables of interest (sick days and wellness visits) because 0 is not contained in our interval and therefore there is evidence that the two have some correlation. This is also supported by our high r value that could be easily computed by the R2 value.

Hypothesis Test

The other option for inference would be to use the p-value to make a judgment on the hypothesis test. In this example, our p-value for the slope is 0.02, which is usually considered significant enough to reject our null hypothesis.

In this instance, our conclusion would be:

Since our p value 0.02<0.05, we reject the null hypothesis. We have significant evidence that the true slope of the regression model between the number of sick days taken and the number of wellness visits is not 0.

Again, since we have some evidence that the slope is not 0, this shows that these two things are correlated, which is also evidenced by the R2 and resulting correlation coefficient .

Example 2: Pick a Test!

(1) A marketing research firm is interested in determining whether the proportion of adults in the United States who use a certain brand of toothpaste is significantly different from 50%. They survey a random sample of 500 adults and find that 270 of them use the toothpaste. Which of the following tests is/are appropriate to use?

(2) A high school statistics teacher wants to determine whether the mean score on a certain statistics exam is significantly different from 80. They administer the exam to a random sample of 25 students and find that the mean score is 78. Which of the following tests is/are appropriate to use?

(3) A psychology researcher is interested in determining whether there is a significant difference in anxiety levels between a treatment group and a control group. They measure anxiety levels in both groups before and after an intervention and find that the mean difference in anxiety levels between the two groups is -5. Which of the following tests is/are appropriate to use?

(4) A political pollster is interested in determining whether the proportion of registered voters who support a certain candidate is significantly different from 40%. They survey a random sample of 1000 registered voters and find that 400 of them support the candidate. They also survey a random sample of 1000 registered voters from a different region and find that 300 of them support the candidate. Which of the following tests is/are appropriate to use?

(5) A nutritionist is interested in determining whether the mean daily caloric intake of a certain population is significantly different from 2000 calories. They collect data from a random sample of 50 individuals from the population and find that the mean caloric intake is 1950 calories. Which of the following tests is/are appropriate to use?

(6) A historian is interested in determining whether the distribution of birth months among a group of people is significantly different from a uniform distribution. They collect data on the birth months of a random sample of 100 people and find that there are more births in the summer months than in the winter months. Which of the following tests is/are appropriate to use?

(7) A sociologist is interested in determining whether there is a significant association between the type of car a person drives and their political party affiliation. They collect data on the car types and political party affiliations of a random sample of 100 people and find that there are more Democrats who drive sedans than Republicans. Which of the following tests is/are appropriate to use?

(8) A medical researcher is interested in determining whether there is a significant difference in the effectiveness of two different treatments for a certain medical condition. They randomly assign patients to receive either treatment A or treatment B and measure the percentage of patients who show improvement after receiving each treatment. They find that the percentage of patients who show improvement is significantly higher for treatment A than for treatment B. Which of the following tests is/are appropriate to use?

(9) A real estate agent is interested in determining whether there is a significant relationship between the size of a house (in square feet) and its sale price. They collect data on the sizes and sale prices of a random sample of houses and find that there is a positive relationship between the two variables. Which of the following tests is/are appropriate to use?

(1) One Proportion Z-Test, One Proportion Z-Interval

(2) One Sample T-Test, One Sample T-Interval

(3) Matched Pairs T-Test

(4) Two Proportion Z-Test, Two Proportion Z-Interval

(5) Two Sample T-Test, Two Sample T-Interval

(6) Chi Squared Goodness of Fit Test

(7) Chi-Squared Test for Independence

(8) Chi-Squared Test for Homogeneity

(9) Linear Regression T-Test, Linear Regression T-Interval

🎥  Watch: AP Stats Unit 9 - Inference for Slopes

Key Terms to Review ( 13 )

Chi-Squared Test for Homogeneity

Chi-Squared Test for Independence

Correlation Coefficient

invT function

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AP Statistics Multiple-Choice Practice Questions: Tests of Significance-Proportions and Means 2

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1. A pharmaceutical company claims that 8% or fewer of the patients taking their new statin drug will have a heart attack in a 5-year period. In a government-sponsored study of 2300 patients taking the new drug, 198 have heart attacks in a 5-year period. Is this strong evidence against the company claim?

2. Is Internet usage different in the Middle East and Latin America? In a random sample of 500 adults in the Middle East, 151 claimed to be regular Internet users, while in a random sample of 1000 adults in Latin America, 345 claimed to be regular users. What is the P -value for the appropriate hypothesis test?

3. What is the probability of mistakenly failing to reject a false null hypothesis when a hypothesis test is being conducted at the 5% significance level (α = 0.05)?

4. A research dermatologist believes that cancers of the head and neck will occur most often of the left side, the side next to a window when a person is driving. In a review of 565 cases of head/neck cancers, 305 occurred on the left side. What is the resulting P -value?

5. Suppose you do five independent tests of the form H 0 : µ = 38 versus H a : µ > 38, each at the α = 0.01 significance level. What is the probability of committing a Type I error and incorrectly rejecting a true null hypothesis with at least one of the five tests?

7. Thirty students volunteer to test which of two strategies for taking multiple-choice exams leads to higher average results. Each student flips a coin, and if heads, uses Strategy A on the first exam and then Strategy B on the second, while if tails, uses Strategy B first and then Strategy A. The average of all 30 Strategy A results is then compared to the average of all 30 Strategy B results. What is the conclusion at the 5% significance level if a two-sample hypothesis test, H 0 : µ 1 = µ 2 , H a : µ 1 ≠ µ 2 , results in a P -value of 0.18?

8. Choosing a smaller level of significance, that is, a smaller α-risk, results in

9. The greater the difference between the null hypothesis claim and the true value of the population parameter,

10. A company selling home appliances claims that the accompanying instruction guides are written at a 6th grade reading level. An English teacher believes that the true figure is higher and with the help of an AP Statistics student runs a hypothesis test. The student randomly picks one page from each of 25 of the company's instruction guides, and the teacher subjects the pages to a standard readability test. The reading levels of the 25 pages are given in the following table:

Assuming that the conditions for inference are met, is there statistical evidence to support the English teacher's belief?

11. Suppose H 0 : p = 0.4, and the power of the test for the alternative hypothesis p = 0.35 is 0.75. Which of the following is a valid conclusion?

12. A factory is located close to a city high school. The manager claims that the plant's smokestacks spew forth an average of no more than 350 pounds of pollution per day. As an AP Statistics project, the class plans a one-sided hypothesis test with a critical value of 375 pounds. Suppose the standard deviation in daily pollution poundage is known to be 150 pounds and the true mean is 385 pounds. If the sample size is 100 days, what is the probability that the class will mistakenly fail to reject the factory manager's false claim?

13. For which of the following is a matched pairs t -test not appropriate?

14. Do high school girls apply to more colleges than high school boys? A two-sample t -test of the hypotheses H 0 : µ girls = µ boys versus H a : µ girls > µ boys results in a P -value of 0.02.

Which of the following statements must be true?

I.A 90% confidence interval for the difference in means contains 0.

II.A 95% confidence interval for the difference in means contains 0.

III.A 99% confidence interval for the difference in means contains 0.

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Power of a Hypothesis Test

The probability of not committing a Type II error is called the power of a hypothesis test.

Effect Size

To compute the power of the test, one offers an alternative view about the "true" value of the population parameter, assuming that the null hypothesis is false. The effect size is the difference between the true value and the value specified in the null hypothesis.

Effect size = True value - Hypothesized value

For example, suppose the null hypothesis states that a population mean is equal to 100. A researcher might ask: What is the probability of rejecting the null hypothesis if the true population mean is equal to 90? In this example, the effect size would be 90 - 100, which equals -10.

Factors That Affect Power

The power of a hypothesis test is affected by three factors.

• Sample size ( n ). Other things being equal, the greater the sample size, the greater the power of the test.
• Significance level (α). The lower the significance level, the lower the power of the test. If you reduce the significance level (e.g., from 0.05 to 0.01), the region of acceptance gets bigger. As a result, you are less likely to reject the null hypothesis. This means you are less likely to reject the null hypothesis when it is false, so you are more likely to make a Type II error. In short, the power of the test is reduced when you reduce the significance level; and vice versa.
• The "true" value of the parameter being tested. The greater the difference between the "true" value of a parameter and the value specified in the null hypothesis, the greater the power of the test. That is, the greater the effect size, the greater the power of the test.

Test Your Understanding

Other things being equal, which of the following actions will reduce the power of a hypothesis test?

I. Increasing sample size. II. Changing the significance level from 0.01 to 0.05. III. Increasing beta, the probability of a Type II error.

(A) I only (B) II only (C) III only (D) All of the above (E) None of the above

The correct answer is (C). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Changing the significance level from 0.01 to 0.05 makes the region of acceptance smaller, which makes the hypothesis test more likely to reject the null hypothesis, thus increasing the power of the test. Since, by definition, power is equal to one minus beta, the power of a test will get smaller as beta gets bigger.

Suppose a researcher conducts an experiment to test a hypothesis. If she doubles her sample size, which of the following will increase?

I. The power of the hypothesis test. II. The effect size of the hypothesis test. III. The probability of making a Type II error.

The correct answer is (A). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Thus, it increases the power of the test. The effect size is not affected by sample size. And the probability of making a Type II error gets smaller, not bigger, as sample size increases.