AP Statistics Sampling Distributions Test

Contents

This ap statistics sampling distributions test checks sampling distributions in 45 minutes, with 8 questions for a total of 20 points.

This free-response test covers sampling distributions for AP Statistics: sample proportions, sample means, the Central Limit Theorem, bias and variability, and differences of two proportions or means. A calculator and z-table are allowed. Check conditions, show your setup, and state conclusions in context. The test has 8 questions worth 20 points and takes about 45 minutes.

Test information

Grade: AP Statistics
Chapter: Sampling Distributions
Common Core: AP Statistics Unit 2 (2026-27)
Time: 45 minutes
Total: 20 points (Question 1: 2 pts; Question 2: 3 pts; Question 3: 2 pts; Question 4: 2 pts; Question 5: 3 pts; Question 6: 3 pts; Question 7: 3 pts; Question 8: 2 pts)

AP Statistics Sampling Distributions Test: questions 1–5

Question 1: Sample proportion of students (2 points)

At a university with \(4{,}000\) students, \(35\%\) have a part-time job. A random sample of \(120\) students is taken, and \(\hat{p}\) is the sample proportion with a part-time job.

Find the mean and standard deviation of the sampling distribution of \(\hat{p}\). (1 point)
Verify that the conditions for the standard deviation formula and a normal model are met. (1 point)

Question 2: Probability of a low proportion (3 points)

A national poll finds that \(45\%\) of adults own a smart speaker. A random sample of \(250\) adults is chosen, and \(\hat{p}\) is the proportion in the sample who own one. The graph shows the sampling distribution of \(\hat{p}\).

Sampling distribution of the sample proportion.

Find the standard deviation of \(\hat{p}\). (1 point)
Find the probability that fewer than \(40\%\) of the sampled adults own a smart speaker. (2 points)

Question 3: Sample size and variability (2 points)

A population has proportion \(p=0.50\). Compare the sampling distributions of \(\hat{p}\) for samples of size \(100\) and size \(400\). Address center and spread, and give the standard deviations. (2 points)

Question 4: Bias and variability (2 points)

A pollster uses three methods to estimate the true proportion \(p=0.40\). The dot plots show \(30\) results from each method; the dashed line is \(p\).

Results of three methods.

Which method has low bias and low variability? Describe the other two. (2 points)

Question 5: Sample mean of wait times (3 points)

Wait times at a clinic are right-skewed with mean \(45\) minutes and standard deviation \(20\) minutes, as shown. A random sample of \(50\) patients is taken (far fewer than \(10\%\) of all patients).

Population of wait times .

Find the mean and standard deviation of the sampling distribution of \(\,\overline{x}\). (1 point)
Describe the shape of the sampling distribution and justify. (1 point)
Find the probability that the sample mean exceeds \(50\) minutes. (1 point)

AP Statistics Sampling Distributions Test: questions 6–8

Question 6: Difference of two proportions (3 points)

In Town 1, \(62\%\) of residents recycle; in Town 2, \(55\%\). Independent random samples of \(150\) residents from Town 1 and \(200\) from Town 2 are taken. Let \(\hat{p}_{1}-\hat{p}_{2}\) be the difference in sample proportions.

Find the mean and standard deviation of \(\hat{p}_{1}-\hat{p}_{2}\). (2 points)
Find the probability that the sample from Town 1 has a lower proportion than the sample from Town 2. (1 point)

Question 7: Difference of two means (3 points)

Delivery times for Service A have mean \(240\) minutes and standard deviation \(30\) minutes; for Service B, mean \(225\) minutes and standard deviation \(25\) minutes. A random sample of \(40\) deliveries from A and \(35\) from B is taken. Let \(\,\overline{x}_{A}-\,\overline{x}_{B}\) be the difference of sample means.

Find the mean and standard deviation of \(\,\overline{x}_{A}-\,\overline{x}_{B}\). (2 points)
Assuming a normal model, find \(P(\,\overline{x}_{A}-\,\overline{x}_{B}> 20)\). (1 point)

Question 8: Is the claim believable? (2 points)

A candidate claims that \(50\%\) of voters support her. In a random sample of \(200\) voters (a small part of all voters), only \(43\%\) support her. Assume the claim is true.