AP Statistics Practice Quiz: Setting Up a Test for the Difference of Two Population Means

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 wants to compare the mean scores of two independent groups of students on a standardized test. Which of the following is the most appropriate method for this analysis?

A)A two-sample t-test for a difference of two population means.B)A one-sample z-test for a population mean.C)A chi-square test for independence.D)A matched-pairs t-test.

All Questions (16)

A researcher wants to compare the mean scores of two independent groups of students on a standardized test. Which of the following is the most appropriate method for this analysis?

A) A two-sample t-test for a difference of two population means.

B) A one-sample z-test for a population mean.

C) A chi-square test for independence.

D) A matched-pairs t-test.

Correct Answer: A

The content states that the appropriate test for a difference of two population means is a two-sample t-test. This scenario involves comparing the means of two distinct, independent groups.

When performing a significance test for the difference between two population means, μ1 and μ2, what is the standard null hypothesis?

A) H0: μ1 - μ2 > 0

B) H0: μ1 = μ2

C) H0: x̄1 - x̄2 = 0

D) H0: μ1 ≠ μ2

Correct Answer: B

The provided content explicitly states that the null hypothesis for a two-sample t-test for a difference of means is H0: μ1 - μ2 = 0, which is equivalent to H0: μ1 = μ2. This hypothesis assumes there is no difference between the population means.

Which of the following are the two primary conditions that must be verified before conducting a significance test for the difference of two population means?

A) The samples must be large, and the population variances must be equal.

B) The data must be paired, and the population distributions must be skewed.

C) The samples must be dependent, and the population standard deviations must be known.

D) The samples must be independent, and the sampling distribution must be approximately normal.

Correct Answer: D

The content specifies two key conditions to check: independence and that the sampling distribution is approximately normal.

An agricultural scientist wants to determine if a new fertilizer (Group 1) yields a different mean crop weight than an old fertilizer (Group 2). Which set of hypotheses is appropriate for this two-sided test?

A) H0: μ1 = μ2; Ha: μ1 > μ2

B) H0: μ1 = μ2; Ha: μ1 ≠ μ2

C) H0: x̄1 = x̄2; Ha: x̄1 ≠ x̄2

D) H0: μ1 - μ2 = 1; Ha: μ1 - μ2 ≠ 1

Correct Answer: B

The null hypothesis posits no difference (H0: μ1 = μ2). Since the scientist wants to know if the yield is 'different' (not specifically greater or less than), a two-sided alternative hypothesis (Ha: μ1 ≠ μ2) is required.

A pharmaceutical company tests a new drug to see if it reduces the mean blood pressure of patients more than a placebo. Let μ1 be the mean reduction for the drug and μ2 be the mean reduction for the placebo. What is the correct alternative hypothesis for this test?

A) Ha: μ1 < μ2

B) Ha: μ1 = μ2

C) Ha: μ1 > μ2

D) Ha: μ1 ≠ μ2

Correct Answer: C

The company wants to know if the new drug's mean reduction (μ1) is greater than the placebo's mean reduction (μ2). This translates to the one-sided alternative hypothesis Ha: μ1 > μ2.

Which of the following is an equivalent way to state the null hypothesis H0: μ1 = μ2?

A) H0: μ1 - μ2 = 0

B) H0: μ1 - μ2 ≠ 0

C) H0: μ1 / μ2 = 0

D) H0: μ1 + μ2 = 0

Correct Answer: A

The content provides two equivalent forms for the null hypothesis: H0: μ1 = μ2 and H0: μ1 - μ2 = 0. Subtracting μ2 from both sides of the first equation gives the second equation.

A teacher compares the final exam scores of her morning class and her afternoon class. She treats the two classes as independent samples. To use a two-sample t-test, she must check that the sampling distribution of the difference in means is approximately normal. Which of the following would satisfy this condition?

A) The sum of the two sample sizes is greater than 30.

B) The two sample standard deviations are nearly equal.

C) Both class sizes are large, or the distribution of scores in both classes is approximately normal.

D) The data was collected from a randomized, controlled experiment.

Correct Answer: C

The condition that the sampling distribution is approximately normal can be met if the underlying population distributions are normal or if the sample sizes are large enough (typically n ≥ 30 for each group) due to the Central Limit Theorem.

A study compares the salaries of employees at two different companies. A random sample of employees is taken from Company A, and a separate random sample is taken from Company B. Why is it important to verify the independence condition for a two-sample t-test in this context?

A) To ensure that the selection of an employee from Company A does not influence the selection of an employee from Company B.

B) To ensure that both companies have the same number of employees.

C) To ensure that the salary distributions are normal.

D) To ensure that the null hypothesis is true.

Correct Answer: A

The independence condition for a two-sample test requires that the two samples are independent of each other. This means the method of selecting one sample should not affect the selection of the other.

The statistical procedure used to test for a significant difference between two population means is called a:

A) Two-proportion z-test

B) One-sample t-test

C) Two-sample t-test

D) Chi-square goodness-of-fit test

Correct Answer: C

The provided content explicitly identifies the appropriate test for a difference of two population means as a two-sample t-test.

The hypotheses for a significance test are always statements about:

A) Sample statistics

B) Sample sizes

C) Population parameters

D) Test conditions

Correct Answer: C

The null and alternative hypotheses are formulated in terms of population parameters (like μ1 and μ2), not sample statistics (like x̄1 and x̄2). The test uses sample statistics to make an inference about the population parameters.

A researcher measures the height of 25 randomly selected male students and 25 randomly selected female students at a university. They wish to perform a two-sample t-test. Which condition for inference has been violated if the researcher actually measured the heights of 25 sets of opposite-sex twins?

A) The normality condition.

B) The 10% condition for sample size.

C) The independence condition.

D) The random sampling condition.

Correct Answer: C

Using twins creates paired or dependent data, as the heights of twins are not independent of each other. A two-sample t-test requires the two samples to be independent.

A quality control manager wants to test if a new production method (Method A) has a lower mean number of defects than the old method (Method B). Which of the following is the correct set of hypotheses?

A) H0: μA = μB; Ha: μA < μB

B) H0: μA = μB; Ha: μA > μB

C) H0: μA = μB; Ha: μA ≠ μB

D) H0: x̄A = x̄B; Ha: x̄A < x̄B

Correct Answer: A

The null hypothesis is that there is no difference between the methods (H0: μA = μB). The manager is testing if the new method is better, meaning it has a 'lower' mean number of defects, which corresponds to a one-sided alternative hypothesis (Ha: μA < μB).

Which of the following scenarios is appropriate for using a two-sample t-test for the difference of two population means?

A) Comparing the proportion of voters who favor a candidate in two different states.

B) Comparing the mean weight of a group of subjects before and after a diet program.

C) Comparing the mean GPAs of a random sample of 50 freshmen and a separate random sample of 50 seniors.

D) Testing if the distribution of car colors in a parking lot matches the manufacturer's stated percentages.

Correct Answer: C

This scenario involves comparing the means of two independent groups (freshmen and seniors). Option A is for proportions, B is for paired data, and D is for a chi-square test.

The null hypothesis H0: μ1 - μ2 = 0 for a two-sample t-test makes an initial assumption that:

A) The two population means are different.

B) The two sample means are equal.

C) The two population means are equal.

D) The two sample sizes are equal.

Correct Answer: C

The null hypothesis is a statement of 'no effect' or 'no difference.' In this context, H0: μ1 - μ2 = 0 directly states that the difference between the two population means is zero, meaning they are assumed to be equal for the purpose of the test.

When setting up a test for the difference of two population means, a researcher selects the two-sample t-test. This selection implies that which of the following is likely true about the populations?

A) The population standard deviations are unknown.

B) The population standard deviations are known.

C) The populations are not independent.

D) The populations are not approximately normal.

Correct Answer: A

The choice of a t-test over a z-test for means is primarily because the population standard deviations (σ) are unknown and must be estimated from the sample data using the sample standard deviations (s). The content specifies a t-test, which is the standard procedure in this common situation.

A city's environmental agency takes water samples from two different rivers, the East River and the West River, to compare the mean pollutant levels. Which of the following represents the correct selection of a testing method and the null hypothesis for this situation?

A) Test: One-sample t-test; H0: μEast = 0

B) Test: Two-sample t-test; H0: μEast > μWest

C) Test: Two-sample z-test for proportions; H0: pEast = pWest

D) Test: Two-sample t-test; H0: μEast - μWest = 0

Correct Answer: D

The situation requires comparing the means of two independent populations, so a two-sample t-test is the appropriate method. The null hypothesis should state that there is no difference between the population means, which can be written as H0: μEast - μWest = 0.