AP Statistics Practice Quiz: Carrying Out a Test for a Population Mean

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

A researcher conducts a significance test for a population mean using a sample of 30 subjects. What are the degrees of freedom for the t-statistic?

A)30B)29C)31D)Cannot be determined from the information given

All Questions (16)

A researcher conducts a significance test for a population mean using a sample of 30 subjects. What are the degrees of freedom for the t-statistic?

A) 30

B) 29

C) 31

D) Cannot be determined from the information given

Correct Answer: B

According to the provided content, the sampling distribution of the t-statistic for a mean has a t-distribution with n-1 degrees of freedom. In this case, n = 30, so the degrees of freedom are 30 - 1 = 29.

In a significance test for a population mean, the calculated p-value is 0.04 and the chosen significance level (alpha) is 0.05. What is the correct formal decision?

A) Reject the null hypothesis because the p-value is less than alpha.

B) Fail to reject the null hypothesis because the p-value is less than alpha.

C) Reject the null hypothesis because the p-value is greater than alpha.

D) Fail to reject the null hypothesis because the p-value is greater than alpha.

Correct Answer: A

The content states that a formal decision compares the p-value to alpha. If the p-value is less than or equal to alpha, the decision is to reject the null hypothesis. Here, 0.04 is less than 0.05, so the null hypothesis is rejected.

The interpretation of a p-value in a significance test for a population mean is conditional on which of the following assumptions?

A) The alternative hypothesis is true.

B) The sample size is large.

C) The null hypothesis is true.

D) The data follows a normal distribution.

Correct Answer: C

As stated in the provided content, interpreting the p-value for a test for a population mean assumes the null hypothesis is true. The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated, given that the null hypothesis is the true state of the population.

A study was conducted to test if a new diet plan reduces weight. The null hypothesis is that the mean weight loss is zero. A p-value of 0.02 was calculated. Which of the following is the correct interpretation of this p-value?

A) There is a 2% chance that the new diet plan is effective.

B) There is a 2% chance of observing the sample results (or more extreme) if the diet plan has no effect on weight.

C) There is a 98% chance that the new diet plan is effective.

D) There is a 2% chance that the null hypothesis is false.

Correct Answer: B

The p-value is the probability of obtaining the observed sample results, or results more extreme, under the assumption that the null hypothesis is true. In this context, the null hypothesis is that the diet has no effect. Therefore, the p-value of 0.02 is the probability of seeing the observed weight loss (or more) if the diet actually has no effect.

A test is conducted to see if the mean response time for a website is less than 3 seconds. From a sample of 20 users, the sample mean is 2.8 seconds with a sample standard deviation of 0.5 seconds. The null hypothesis is H₀: μ = 3. Which of the following shows the correct calculation for the test statistic?

A) t = (2.8 - 3) / (0.5 / √20)

B) t = (3 - 2.8) / (0.5 / √20)

C) t = (2.8 - 3) / 0.5

D) t = (2.8 - 3) / (0.5 / 20)

Correct Answer: A

The content requires the ability to calculate an appropriate test statistic. The formula for the t-statistic is (sample mean - hypothesized population mean) / (sample standard deviation / √n). Plugging in the values gives t = (2.8 - 3) / (0.5 / √20).

A significance test for the mean caffeine content of a soda brand results in a p-value of 0.18. The company claimed the mean was 40 mg. Which of the following is the most appropriate justification of the claim based on this result at a 5% significance level?

A) We have strong evidence that the mean caffeine content is 40 mg.

B) We have proven that the mean caffeine content is not 40 mg.

C) We fail to reject the null hypothesis; there is not sufficient evidence to conclude the mean caffeine content is different from 40 mg.

D) We reject the null hypothesis; there is sufficient evidence to conclude the mean caffeine content is different from 40 mg.

Correct Answer: C

The p-value (0.18) is greater than the significance level (0.05), so we fail to reject the null hypothesis. The content requires justifying a claim based on the test results. The correct justification is to state the decision (fail to reject) and conclude that there is not enough statistical evidence to support the alternative hypothesis (that the mean is different from 40 mg).

A matched pairs experiment is designed to test if a new fertilizer increases crop yield. Data is collected from 18 plots of land, with each plot split into two subplots (one with the new fertilizer, one without). What is the appropriate test statistic and its corresponding degrees of freedom?

A) A t-statistic with 17 degrees of freedom.

B) A t-statistic with 18 degrees of freedom.

C) A t-statistic with 35 degrees of freedom.

D) A z-statistic with 17 degrees of freedom.

Correct Answer: A

For a matched pairs design, we perform a one-sample t-test on the differences. The sample size, n, is the number of pairs. Here, n = 18. The test statistic is a t-statistic, and its distribution has n-1 degrees of freedom. Therefore, the correct degrees of freedom are 18 - 1 = 17.

How do the results of a significance test for a mean contribute to answering a research question?

A) They prove the research hypothesis is definitively true or false.

B) They provide a measure of the practical importance of the findings.

C) They offer statistical reasoning to support or fail to support a claim about a population mean.

D) They describe the characteristics of the sample data collected.

Correct Answer: C

The content states that the results of a significance test for a mean provide statistical reasoning for a research question. The test allows us to use sample data to make an inference and justify a claim about the larger population, which is the goal of the research question.

In a matched pairs test, a p-value of 0.03 is found for the alternative hypothesis Hₐ: μ_d > 0, where μ_d is the mean of the differences. What is the correct interpretation?

A) There is a 3% probability that the mean of the differences is greater than 0.

B) If the true mean of the differences were 0, there would be a 3% probability of observing a sample mean difference as large or larger than the one obtained.

C) There is a 97% probability that the true mean of the differences is 0.

D) The test proves that the mean of the differences is greater than 0.

Correct Answer: B

The content requires the ability to interpret a p-value for a matched pairs test. The p-value is the probability of getting the observed result (or one more extreme) assuming the null hypothesis (H₀: μ_d = 0) is true. Option B correctly states this interpretation in the context of the problem.

A significance test is performed to determine if the mean number of hours students study per week is different from 15 hours. A p-value of 0.008 is obtained. Using a significance level of α = 0.01, what is the correct conclusion?

A) Since 0.008 < 0.01, we fail to reject H₀. There is not enough evidence to say the mean is different from 15.

B) Since 0.008 > 0.01, we reject H₀. There is enough evidence to say the mean is different from 15.

C) Since 0.008 < 0.01, we reject H₀. There is sufficient evidence to conclude the mean study time is different from 15 hours.

D) Since 0.008 < 0.01, we accept H₀. We conclude the mean study time is 15 hours.

Correct Answer: C

The formal decision rule is to compare the p-value to alpha. Here, p-value (0.008) is less than alpha (0.01), so we reject the null hypothesis. The content requires justifying a claim based on this result. The correct justification is to state that there is sufficient evidence for the alternative hypothesis, which is that the mean study time is different from 15 hours.

When conducting a significance test for a population mean, the sampling distribution of the calculated test statistic follows which distribution, provided the population standard deviation is unknown?

A) A Normal distribution

B) A Binomial distribution

C) A t-distribution

D) A Chi-squared distribution

Correct Answer: C

The provided content explicitly states that the sampling distribution of the t-statistic for a mean has a t-distribution with n-1 degrees of freedom.

A researcher tests if a new teaching method improves test scores. A matched pairs design is used on 25 students, testing them before and after the new method. The mean of the differences (after - before) was 5 points with a standard deviation of the differences of 10 points. What is the calculated test statistic for this test?

A) t = (5 - 0) / (10 / √24)

B) t = (5 - 0) / 10

C) t = (5 - 0) / (10 / √25)

D) t = 5 / (10 * √25)

Correct Answer: C

For a matched pairs test, a one-sample t-test is performed on the differences. The null hypothesis is that the mean difference is 0. The test statistic is t = (sample mean difference - hypothesized mean difference) / (sample standard deviation of differences / √n). Here, the sample mean difference is 5, the hypothesized mean difference is 0, the standard deviation of the differences is 10, and n (the number of pairs) is 25. The correct calculation is t = (5 - 0) / (10 / √25).

A formal decision in a significance test for a mean is made by comparing which two values?

A) The sample mean and the population mean

B) The p-value and the significance level (alpha)

C) The test statistic and the critical value

D) The sample size and the degrees of freedom

Correct Answer: B

The content specifies that a formal decision compares the p-value to alpha to determine whether to reject or fail to reject the null hypothesis.

A test of H₀: μ = 100 versus Hₐ: μ ≠ 100 results in a p-value of 0.34. An appropriate justification based on this result would be:

A) The result is not statistically significant, providing evidence that the true population mean is 100.

B) The result is not statistically significant, indicating that we do not have sufficient evidence to conclude the true population mean is different from 100.

C) The result is statistically significant, indicating that we have sufficient evidence to conclude the true population mean is different from 100.

D) The result is statistically significant, providing evidence that the true population mean is 100.

Correct Answer: B

With a p-value of 0.34, which is higher than any common alpha level (e.g., 0.01, 0.05, 0.10), we fail to reject the null hypothesis. This means the result is not statistically significant. Justifying the claim involves stating that we lack sufficient evidence for the alternative hypothesis. Failing to reject H₀ does not prove H₀ is true, it only means we don't have enough evidence to say it's false.

A matched pairs study involving 12 participants is conducted to see if a meditation app reduces stress levels. A t-test is performed on the differences in stress scores. What are the degrees of freedom for the appropriate t-distribution?

A) 12

B) 11

C) 23

D) 24

Correct Answer: B

In a matched pairs study, the sample size 'n' is the number of pairs or participants. The degrees of freedom for the t-statistic are n-1. With 12 participants, n=12, so the degrees of freedom are 12 - 1 = 11.

A researcher wants to provide statistical reasoning to answer the question, 'Is the mean cholesterol level for a certain group of patients above 200 mg/dL?' After collecting data and performing a significance test, a p-value of 0.015 is calculated. How does this result address the research question?

A) It proves that the mean cholesterol level is exactly 200 mg/dL.

B) It suggests, with a low probability of being wrong due to random chance alone, that there is evidence the mean cholesterol level is above 200 mg/dL, assuming a standard alpha level.

C) It shows that 1.5% of the patients have a cholesterol level above 200 mg/dL.

D) It provides no reasoning because the p-value is a probability, not a direct answer to the question.

Correct Answer: B

The content states that a significance test provides statistical reasoning for a research question. A low p-value (like 0.015) indicates that the observed sample result is unlikely if the null hypothesis (mean = 200) were true. This provides statistically significant evidence for the alternative hypothesis (mean > 200), thus providing a reasoned answer to the research question.