AP Statistics Practice Quiz: Confidence Intervals for the Difference of Two Proportions

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 proportion of students who pass a certification exam from two different training programs. Which is the appropriate statistical procedure to estimate the difference between these two population proportions?

A)A two-sample z-interval for a difference between proportionsB)A two-sample t-interval for a difference between meansC)A one-sample z-interval for a proportionD)A chi-square test for independence

All Questions (16)

A) A two-sample z-interval for a difference between proportions

B) A two-sample t-interval for a difference between means

C) A one-sample z-interval for a proportion

D) A chi-square test for independence

Correct Answer: A

The provided content states that the appropriate procedure for a two-sample comparison of proportions is a two-sample z-interval. This scenario involves comparing two proportions from two independent groups.

Before calculating a confidence interval for the difference between two population proportions, which two conditions must be verified?

A) Randomness and Linearity

B) Independence and Approximate Normality

C) Constant Variance and Randomness

D) Large Sample Size and Linearity

Correct Answer: B

The provided content explicitly states that to calculate a confidence interval for a difference between proportions, one must check for independence and approximate normality.

In the formula for a confidence interval for the difference of two proportions, (p1-hat - p2-hat) ± z* * SE, what does the term (p1-hat - p2-hat) represent?

A) The margin of error

B) The standard error

C) The point estimate for the difference

D) The critical value

Correct Answer: C

The value (p1-hat - p2-hat) is the difference between the two sample proportions, which serves as the best single-value estimate, or point estimate, for the true difference in population proportions.

A 95% confidence interval for the difference in the proportion of satisfied customers between Company A and Company B (pA - pB) is calculated to be (0.05, 0.15). How is this interval estimate interpreted?

A) We are 95% confident that the proportion of satisfied customers for Company A is between 5% and 15%.

B) We are 95% confident that the true difference in the proportion of satisfied customers (pA - pB) is between 0.05 and 0.15.

C) There is a 95% probability that the sample difference will be between 0.05 and 0.15.

D) The difference in the sample proportions is exactly 10%.

Correct Answer: B

The content states that confidence intervals for a difference in proportions can be used to calculate interval estimates. The interval (0.05, 0.15) provides a range of plausible values for the true difference between the two population proportions.

A city council wants to determine if there is a difference in the proportion of residents who support a new park in District 1 versus District 2. They take random samples from each district. Which procedure should they use to construct a confidence interval for this comparison?

A) A two-sample z-interval for a difference of proportions

B) A one-sample z-interval for the overall proportion of support

C) A two-sample t-interval for the difference of means

D) A matched-pairs t-interval

Correct Answer: A

The goal is to compare two population proportions from two independent samples. The content identifies the two-sample z-interval as the appropriate procedure for such a comparison.

When constructing a two-sample z-interval for the difference of proportions, why is it necessary to check for approximate normality?

A) To ensure the samples are independent of each other.

B) To justify the use of the z* critical value from the standard normal distribution.

C) To confirm that the population proportions are equal.

D) To calculate the point estimate (p1-hat - p2-hat).

Correct Answer: B

The procedure is a 'z-interval,' which relies on the standard normal (z) distribution. The condition of approximate normality ensures that the sampling distribution of the difference in proportions is close enough to a normal distribution to make the use of the z* critical value valid.

The general formula for a confidence interval is Point Estimate ± Margin of Error. For a two-sample z-interval for the difference of proportions, which part of the formula (p1-hat - p2-hat) ± z* * SE represents the margin of error?

A) (p1-hat - p2-hat)

B) z*

C) SE

D) z* * SE

Correct Answer: D

The margin of error is the value that is added to and subtracted from the point estimate to create the interval. Based on the provided formula, the margin of error is the product of the critical value (z*) and the standard error (SE).

A confidence interval for the difference between the proportion of men and women who prefer a certain brand (p_men - p_women) is found to be (-0.08, 0.12). What is the point estimate (p_men-hat - p_women-hat) used to construct this interval?

A) 0.02

B) 0.04

C) 0.10

D) 0.20

Correct Answer: A

The point estimate is the center of the confidence interval. It can be calculated by averaging the endpoints: (-0.08 + 0.12) / 2 = 0.04 / 2 = 0.02.

A researcher is comparing data from two independent groups. Which of the following indicates that a two-sample z-interval for a difference of proportions is the appropriate procedure, rather than a two-sample t-interval?

A) The data involves comparing the means of the two groups.

B) The data is categorical and involves comparing the proportions of the two groups.

C) The population standard deviation is unknown.

D) The sample sizes are small.

Correct Answer: B

The content specifies that the two-sample z-interval is used for a 'comparison of population proportions.' A t-interval is used for comparing means. Therefore, the categorical nature of the data (leading to proportions) is the key indicator.

When verifying conditions for a two-sample z-interval for a difference of proportions, the independence condition must be checked. What does this condition generally refer to?

A) The two sample proportions, p1-hat and p2-hat, must be exactly equal.

B) The observations within each sample and between the two samples should be independent.

C) The sampling distribution must be approximately normal.

D) The sample sizes must be greater than 30.

Correct Answer: B

The independence condition is a required check for this procedure. It requires that the selection of individuals for one group does not influence the selection for the other group, and that individuals within each group are also independent of one another.

A study aims to estimate the difference in the proportion of adults who use social media in two different countries. The point estimate for the difference is 0.11, and the margin of error for a 90% confidence level is 0.04. What is the resulting 90% confidence interval?

A) (0.07, 0.11)

B) (0.11, 0.15)

C) (0.07, 0.15)

D) (0.09, 0.13)

Correct Answer: C

The interval is calculated as (point estimate) ± (margin of error). Using the given values, the interval is 0.11 ± 0.04, which results in a lower bound of 0.11 - 0.04 = 0.07 and an upper bound of 0.11 + 0.04 = 0.15. The interval is (0.07, 0.15).

Suppose a confidence interval for the difference in the proportion of defective parts from two suppliers (p_supplier1 - p_supplier2) is (-0.03, 0.01). Which statement correctly interprets this interval estimate?

A) The difference in the number of defective parts is between -3 and 1.

B) We are confident that the proportion of defective parts for Supplier 1 is between 1% lower and 3% higher than for Supplier 2.

C) We are confident that the true difference in the proportion of defective parts is between 1% lower and 3% higher than the sample difference.

D) We are confident that the true difference in the proportion of defective parts (p_supplier1 - p_supplier2) is between -0.03 and 0.01.

Correct Answer: D

The content states that these intervals are used to calculate an interval estimate for a difference of proportions. The interval (-0.03, 0.01) provides a range of plausible values for the true difference in population proportions, which may be interpreted in the context of the problem's units (in this case, the proportion of defective parts).

A 99% confidence interval for the difference in the proportion of voters favoring a candidate in two states (p_stateA - p_stateB) is (-0.04, 0.06). Based on this interval, is there convincing evidence of a difference in the proportion of voters favoring the candidate between the two states?

A) Yes, because the interval is very narrow.

B) Yes, because the point estimate is positive.

C) No, because the interval contains 0.

D) No, because the confidence level is too high.

Correct Answer: C

The interval estimate for the true difference in proportions is (-0.04, 0.06). Since this interval includes the value 0, it is plausible that there is no difference between the two population proportions (p_stateA - p_stateB = 0). Therefore, there is not convincing evidence of a difference.

Which of the following correctly represents the structure for calculating an interval estimate for a comparison of proportions?

A) (p1-hat - p2-hat) ± t* * SE

B) (x-bar1 - x-bar2) ± z* * SE

C) (p1-hat - p2-hat) ± z* * SE

D) p-hat ± z* * SE

Correct Answer: C

The provided content explicitly gives the formula for the interval estimate for a comparison of proportions as (p1-hat - p2-hat) ± z* * SE.

A researcher calculates a 95% confidence interval for the difference in proportions of successful treatments between a new drug and a placebo (p_drug - p_placebo) as (0.08, 0.20). What was the margin of error for this interval?

A) 0.06

B) 0.08

C) 0.12

D) 0.14

Correct Answer: A

The margin of error is half the width of the confidence interval. The width is 0.20 - 0.08 = 0.12. Therefore, the margin of error is 0.12 / 2 = 0.06.

What is the primary goal of constructing a two-sample z-interval for the difference between population proportions?

A) To prove that the two population proportions are different.

B) To provide a point estimate for the difference between the two sample proportions.

C) To provide a range of plausible values for the true difference between the two population proportions.

D) To determine if the samples were collected randomly.

Correct Answer: C

The content states that the procedure is used to 'calculate an appropriate confidence interval for a comparison of population proportions' and to 'calculate an interval estimate.' The purpose of any confidence interval is to provide an interval estimate—a range of plausible values—for an unknown population parameter, which in this case is the difference between two population proportions.