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Sampling Distributions Test – AP Statistics Math (Printable PDF)

Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

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This 45-minute test has eight AP-style questions worth 20 points. State and check conditions, show the standardizing step, and interpret results in context.

Standards: AP Statistics Unit 2 (2026-27) · Level: Grade 12 AP Statistics

Time: 45 minutes · 8 questions · Total: 20 points · Calculator: as allowed by your teacher

Question 1 Parameter and Statistic (2 points)

A school wants the mean commute time of all its 1,400 students. A random sample of 60 students has a mean of 24.5 minutes. (a) Identify the parameter and the statistic, with symbols. (b) Would another random sample of 60 students be expected to have exactly the same mean? Explain.

Question 2 Sampling Distribution of p̂ (3 points)

In a state, 52% of voters support a ballot measure. A random sample of 180 voters is taken. (a) Verify that the sampling distribution of p̂ is approximately normal. (b) Find its mean and standard deviation. (c) Find the probability that p̂ is less than 0.47.

Question 3 Sample Size and Spread (2 points)

The figure shows the sampling distributions of p̂ for samples of size 25 and size 100 from a population with p = 0.5. The narrow curve has the smaller spread.

0.30.40.50.60.7Sample proportion
Two sampling distributions of p̂.
  1. Which sample size produced the narrow curve? Explain. (1 point)
  2. Find the standard deviation of p̂ for each sample size. (1 point)

Question 4 Sample Mean from a Normal Population (3 points)

The lengths of steel rods are approximately normal with mean 125 mm and standard deviation 6 mm. A random sample of 9 rods is selected. (a) Find the mean and standard deviation of x̄. (b) Justify using a normal model for x̄. (c) Find P(122 < x̄ < 128).

Question 5 Central Limit Theorem Shape (2 points)

The histogram shows a skewed right population.

010203040Percent01020304050Repair cost (hundreds of dollars)
Repair costs for a population of appliances.
  1. Describe the shape of the sampling distribution of x̄ for samples of size 4. (1 point)
  2. Describe the shape for samples of size 50. Explain. (1 point)

Question 6 Difference of Proportions (3 points)

In district A, 60% of parents favor a later start time; in district B, 50% do. Independent random samples of 100 parents from A and 120 from B are taken. (a) Find the mean and standard deviation of p̂A − p̂B. (b) Find the probability that p̂A < p̂B, assuming the sampling distribution is approximately normal.

Question 7 Difference of Means (2 points)

Delivery times for company A have mean 30 minutes and standard deviation 5; for company B, mean 27 minutes and standard deviation 4. Independent random samples of 25 deliveries from A and 36 from B are taken. Find the mean and standard deviation of x̄A − x̄B.

Question 8 Bias and Variability (3 points)

Two methods estimate a population proportion whose true value is 0.40. Each dotplot shows 10 estimates.

0.20.30.40.50.6Estimates from method A
Method A (true parameter 0.40).
0.20.30.40.50.6Estimates from method B
Method B (true parameter 0.40).
  1. Describe the bias and variability of method A. (1 point)
  2. Describe the bias and variability of method B. (1 point)
  3. Which method would you choose, and why? (1 point)

View the Sampling Distributions test answer key with scoring

Sampling Distributions lessonSampling Distributions worksheetSampling Distributions quiz

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