Sampling Distributions: AP Statistics Math Test – PDF Download

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.

1. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\). (1 point)
2. 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 \(\hat{p}\).

1. Find the standard deviation of \(\hat{p}\). (1 point)
2. 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 \(p\).

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).

1. Find the mean and standard deviation of the sampling distribution of \(\,\overline{x}\). (1 point)
2. Describe the shape of the sampling distribution and justify. (1 point)
3. 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.

1. Find the mean and standard deviation of \(\hat{p}_{1}-\hat{p}_{2}\). (2 points)
2. 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.

1. Find the mean and standard deviation of \(\,\overline{x}_{A}-\,\overline{x}_{B}\). (2 points)
2. 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.

1. Find the probability of a sample proportion of \(0.43\) or less. (1 point)
2. What does this result suggest about the claim? (1 point)

How to use this ap statistics sampling distributions test

Take this ap statistics sampling distributions test without notes, then grade it with the test answer key out of 20 points.

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