AP Statistics Practice Quiz: Sampling Distributions for Differences in Sample 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

According to the properties of sampling distributions, what is the mean of the sampling distribution of the difference in sample proportions, p̂1 - p̂2?

A)p1 + p2B)p1 - p2C)The average of the two sample sizes, (n1+n2)/2D)It cannot be determined without the sample data.

All Questions (16)

According to the properties of sampling distributions, what is the mean of the sampling distribution of the difference in sample proportions, p̂1 - p̂2?

A) p1 + p2

B) p1 - p2

C) The average of the two sample sizes, (n1+n2)/2

D) It cannot be determined without the sample data.

Correct Answer: B

The sampling distribution of the difference in sample proportions (p̂1 - p̂2) is centered at the true difference in population proportions, which is p1 - p2. This is a fundamental parameter of the distribution.

Which condition must be met for the sampling distribution of a difference in sample proportions to be described as approximately normal?

A) The sample sizes must be equal.

B) The population proportions must be equal.

C) The Large Counts Condition must be met for both samples.

D) The samples must be taken with replacement.

Correct Answer: C

The shape of the sampling distribution of the difference in sample proportions is approximately normal if the Large Counts Condition (n*p ≥ 10 and n*(1-p) ≥ 10) is satisfied for both of the individual samples.

A researcher is comparing the proportion of residents in City A who own a pet to the proportion in City B. They take a random sample of 200 residents from City A (population 500,000) and 150 from City B (population 300,000). When calculating the standard deviation of the difference in sample proportions, what condition allows for the use of the standard deviation formula?

A) The samples are random.

B) The sample sizes are less than 10% of their respective population sizes.

C) The Large Counts Condition is met for both samples.

D) The population sizes are greater than 1000.

Correct Answer: B

When sampling without replacement, the formula for the standard deviation of the sampling distribution is approximately correct as long as each sample size is no more than 10% of its population size. This is known as the 10% condition, which ensures the independence of observations.

Let p1 be the true proportion of high school seniors who attend their prom and p2 be the true proportion of juniors who attend. The mean of the sampling distribution for the difference in sample proportions is p1 - p2 = 0.25. Which of the following is the best interpretation of this parameter?

A) In any pair of samples, the proportion of seniors attending prom will be 0.25 higher than the proportion of juniors.

B) There is a 25% chance that a randomly selected senior will attend prom and a junior will not.

C) Over many pairs of random samples of the same size, the average difference between the sample proportion of seniors and juniors attending prom is expected to be 0.25.

D) The probability of observing a difference of exactly 0.25 in the sample proportions is very high.

Correct Answer: C

The mean of a sampling distribution represents the long-run average of the statistic over many, many samples. Therefore, p1 - p2 = 0.25 means the expected average difference between the sample proportions (p̂1 - p̂2) is 0.25.

A researcher wants to compare the proportion of students who pass a statistics final at two different colleges. At College 1, the true pass rate is p1 = 0.80, and at College 2, it is p2 = 0.70. Samples of n1 = 40 from College 1 and n2 = 50 from College 2 are taken. Can the sampling distribution of the difference in sample proportions be described as approximately normal?

A) Yes, because both sample sizes are greater than 30.

B) Yes, because the Large Counts Condition is met for both samples.

C) No, because the Large Counts Condition is not met for College 1.

D) No, because the Large Counts Condition is not met for College 2.

Correct Answer: C

To check for normality, we must verify the Large Counts Condition for both samples. For College 1: n1*p1 = 40*0.80 = 32 (≥10), but n1*(1-p1) = 40*0.20 = 8 (<10). Since one of these values is less than 10, the condition is not met for the first sample, and the distribution cannot be assumed to be approximately normal.

Which of the following correctly identifies the parameters of the sampling distribution for a difference in sample proportions?

A) A mean of p̂1 - p̂2 and a specific standard deviation.

B) A mean of 0 and a standard deviation of 1.

C) A mean of p1 - p2 and a specific standard deviation.

D) A mean based on sample sizes and a standard deviation of p1 - p2.

Correct Answer: C

Parameters describe the population or the true sampling distribution, not the sample statistics. The mean of the sampling distribution is the true difference in population proportions, p1 - p2, and it has a specific, defined standard deviation based on the population parameters.

Suppose that for the difference in proportions of voters favoring a policy in two districts, the probability of observing a difference (p̂1 - p̂2) of -0.05 or less is calculated to be 0.18. What is the correct interpretation of this value?

A) There is an 18% chance that the true difference in proportions is -0.05.

B) If we take many pairs of samples, about 18% of them will result in the sample proportion from District 1 being at least 5 percentage points lower than that from District 2.

C) The sample proportion from District 1 is 18% likely to be lower than the sample proportion from District 2.

D) In 18% of all possible samples, the policy is less popular in District 1.

Correct Answer: B

A probability calculated from a sampling distribution refers to the likelihood of observing a certain sample statistic (or one more extreme). A difference of -0.05 means p̂1 is 0.05 less than p̂2. The probability of 0.18 applies to this outcome occurring over repeated sampling.

A student is checking conditions to model the sampling distribution of p̂A - p̂B. They correctly verify that nA*pA ≥ 10 and nA*(1-pA) ≥ 10. They then conclude the distribution is approximately normal. Why might this conclusion be premature?

A) They did not check if the samples were random.

B) They did not check the Large Counts Condition for the second sample from population B.

C) They did not calculate the mean of the distribution first.

D) They did not verify that the sample sizes were less than 10% of the population sizes.

Correct Answer: B

For the sampling distribution of a *difference* in proportions to be approximately normal, the Large Counts Condition must be met for *both* samples independently. Checking only one sample is insufficient to justify the normality assumption for the distribution of the difference.

A pharmaceutical company is testing a new drug. The true proportion of patients experiencing relief is p_drug = 0.75, while for a placebo it is p_placebo = 0.50. Random samples of 100 patients are assigned to each group. The populations of potential patients are very large. Which statement best describes the sampling distribution of the difference in sample proportions (p̂_drug - p̂_placebo)?

A) Approximately normal with a mean of 0.25.

B) Approximately normal with a mean of 0.

C) Not approximately normal because the proportions are different.

D) Shape is unknown, but the mean is 0.25.

Correct Answer: A

First, the mean is p1 - p2 = 0.75 - 0.50 = 0.25. Second, we check the Large Counts Condition for both samples. Drug: 100*0.75=75 and 100*0.25=25 (both ≥10). Placebo: 100*0.50=50 and 100*0.50=50 (both ≥10). Since the condition is met for both, the distribution is approximately normal with a mean of 0.25.

The interpretation of parameters for a sampling distribution for a difference of proportions should always be done in context. Why is this important?

A) Because context is required to calculate the standard deviation.

B) Because context makes the numbers more relatable and provides practical meaning to the results.

C) Because context determines whether the Large Counts Condition is met.

D) Because context is the only way to verify that the samples were random.

Correct Answer: B

While calculations are purely mathematical, their interpretation is meaningless without context. Stating that the mean difference is 0.10 is less informative than stating that, on average, the proportion of satisfied customers for Product A is expected to be 10 percentage points higher than for Product B. Context provides the 'so what' for the analysis.

A school district has 4,000 high school students and 3,000 middle school students. A researcher samples 500 high schoolers and 400 middle schoolers to compare the proportion who participate in extracurricular activities. Why is the standard deviation formula for the difference in sample proportions likely to be inaccurate in this case?

A) The sample sizes are too large.

B) The population proportions are unknown.

C) At least one sample size is more than 10% of its population size.

D) The Large Counts Condition is unlikely to be met.

Correct Answer: C

The 10% condition is required for the standard deviation formula to be accurate when sampling without replacement. High school sample: 500/4000 = 12.5%. Middle school sample: 400/3000 = 13.3%. Since both samples are larger than 10% of their respective populations, the independence assumption is violated, and the standard deviation formula is not appropriate.

To determine the parameters of a sampling distribution for a difference in proportions, what information is required?

A) The two sample proportions (p̂1 and p̂2) and the two sample sizes (n1 and n2).

B) The two population proportions (p1 and p2) and the two sample sizes (n1 and n2).

C) Only the two sample sizes (n1 and n2).

D) Only the two population proportions (p1 and p2).

Correct Answer: B

Parameters describe the theoretical sampling distribution, which depends on the true population proportions (p1 and p2) and the sizes of the samples (n1 and n2) being taken. The mean is p1 - p2, and the standard deviation formula also uses these four values.

For a study comparing the proportion of men (p1) and women (p2) who prefer coffee over tea, samples of n1=60 men and n2=50 women are taken. The Large Counts Condition is met if:

A) The total number of successes and failures in the combined sample are both at least 10.

B) n1*p1, n1*(1-p1), n2*p2, and n2*(1-p2) are all at least 10.

C) n1*p2 and n2*p1 are both at least 10.

D) The average of n1 and n2 is greater than 30.

Correct Answer: B

The Large Counts Condition must be checked separately for each sample. This means checking that the number of successes (n*p) and failures (n*(1-p)) are both at least 10 within the male sample and also within the female sample.

Before calculating a probability using the normal model for the sampling distribution of p̂1 - p̂2, which of the following is the most complete list of items to check and determine?

A) Only the mean and standard deviation of the distribution.

B) Only that the Large Counts Condition is met for both samples.

C) The mean of the distribution and whether the 10% condition is met for both samples.

D) The parameters (mean and standard deviation) of the distribution and whether the conditions for approximate normality (Large Counts) and independence (10% condition) are met.

Correct Answer: D

To perform a valid probability calculation, you need a complete picture of the sampling distribution. This involves determining its parameters (mean and standard deviation) and verifying the conditions that justify your choice of model (approximate normality via Large Counts) and the formula for the standard deviation (independence via the 10% condition).

If the true proportion of teenagers who own a smartphone is p1 = 0.90 and the true proportion of adults (age 65+) who own one is p2 = 0.60, what is the expected value of the difference in sample proportions, p̂1 - p̂2?

A) 0.30

B) 1.50

C) 0.75

D) It depends on the sample sizes.

Correct Answer: A

The expected value of the difference in sample proportions is the mean of its sampling distribution. The mean of the sampling distribution of p̂1 - p̂2 is equal to the true difference in population proportions, p1 - p2. Here, 0.90 - 0.60 = 0.30.

A political analyst compares the approval rating of a governor in two consecutive months. In month 1, the true approval was p1=0.52 and in month 2 it was p2=0.48. A polling organization surveys 1,000 different voters each month. The sampling distribution of the difference in sample proportions (p̂1 - p̂2) is approximately normal. What does this normality allow the analyst to do?

A) Conclude that the governor's approval rating has definitely dropped.

B) Calculate the exact difference in approval for the two months.

C) Calculate the probability of observing a certain difference in sample proportions (or a more extreme one) by chance.

D) Prove that the samples were collected randomly.

Correct Answer: C

The primary benefit of knowing that a sampling distribution is approximately normal is that it allows us to use the normal distribution (and z-scores) to calculate probabilities for sample outcomes. This helps determine if an observed difference is statistically significant or likely due to random sampling variability.