AP Statistics Unit 5: Sampling Distributions

Unit 5 covers sampling distributions: what happens to a statistic across repeated samples, and the Central Limit Theorem. It is the bridge between probability and inference, and it is where students most often lose the thread of the course.

Practise this unit freeAll Statistics units
70 questions7 CED topics100% with a figure
How this unit behaves

What 70 questions on it look like

100% of the questions come with a figure — a graph, diagram or data display. Reading the figure correctly is most of the work before any content knowledge applies.

Sampling Distributions for Sample Proportions49
Sampling Distributions for Sample Means5
Biased and Unbiased Point Estimates5
Sampling Distributions for Differences in Sample Proportions4
Sampling Distributions for Differences in Sample Means3
The Central Limit Theorem3
The Normal Distribution, Revisited1

Questions per CED topic.

From the bank

Three real sampling distributions questions

Drawn from the practice pool, not from the mock papers.

Sampling Distributions for Sample Proportions · Statistical Argumentation · with figure

For random samples of size 180, suppose p = 0.09 for sample boxes converted to orders. Which check addresses whether p-hat is approximately normal?

  1. AOnly 180 > 30 must be checked.
  2. BUse sqrt(p(1-p))
  3. COnly p = 0.09 must be checked.
  4. D180(0.09) = 16.2 and 180(0.91) = 163.8correct
  5. EThe check requires exactly 10 observed successes.
Why D is correct

Check np and n(1-p): 180(0.09) = 16.2 and 180(0.91) = 163.8.

large-count check for sample boxes converted to orders: in the "For random samples of size 180, suppose p = 0.09 for sample boxes converted to orders. Which check addresses whether p-hat is approximately normal?" setting, especially sample boxes converted to orders; trial conversions, identify the sampling-distribution center, spread, and conditions before using a model. The professional check is to keep the values 180, 0.09, the sample proportion normal condition method, and the final answer choice aligned with Sampling Distributions.

Common mistakes. For the stem cue "For random samples of size one hundred eighty, suppose p = zero point zero nine for sample boxes converted to orders. Which check addresses whether p-hat is approximately normal?", a likely mistake is for proportions, expected success and failure counts are needed. Keep the calculation or conclusion tied to the original context.

Biased and Unbiased Point Estimates · Statistical Argumentation · with figure

The distribution of weekly grocery spending for individual households in a city is strongly right-skewed, with mean mu and standard deviation sigma. For random samples of 64 households, what can be said about the sampling distribution of the sample mean spending?

  1. AIt is approximately normal, with mean mu and standard deviation sigma/sqrt(64).correct
  2. BIt has unknown shape because the population distribution is not normal.
  3. CThe population standard deviation is also the sampling-distribution spread, with no adjustment for the sample size.
  4. DIt is strongly right-skewed, with mean mu and standard deviation sigma.
  5. EIt is approximately normal, with mean 64mu and standard deviation 64sigma.
Why A is correct

Because the samples are random and the sample size is large, the central limit theorem says the sampling distribution of the sample mean is approximately normal, with mean mu and standard deviation sigma/sqrt(64).

Applying the central limit theorem to sample means: in the "The distribution of weekly grocery spending for individual households in a city is strongly right-skewed, with mean mu and standard deviation sigma. For random samples of 64 households, what can be said about the sampling distribution of the sample mean spending?" setting, especially strongly right skewed; weekly grocery spending, identify the sampling-distribution center, spread, and conditions before using a model. The professional check is to keep the values 64, the sampling distribution mean method, and the final answer choice aligned with Sampling Distributions.

Common mistakes. For the stem cue "The distribution of weekly grocery spending for individual households in a city is strongly right-skewed, with mean mu and standard deviation sigma. For random samples of sixty four households, what can be", a likely mistake is this describes the individual household distribution, not the sampling distribution of the mean. Keep the calculation or conclusion tied to the original context.

The Normal Distribution, Revisited · Using Probability and Simulation · with figure

Suppose the true proportion of residents in a town who recycle batteries is p. Two sampling plans take independent random samples, one with n = 25 residents and one with n = 100 residents. How does the standard deviation of the sample proportion for n = 100 compare with that for n = 25?

  1. AIt is twice as large.
  2. BIt is half as large.correct
  3. CIt is the same size because p has not changed.
  4. DIt is one-fourth as large.
  5. EUse population spread.
Why B is correct

The standard deviation of a sample proportion is sqrt(p(1-p)/n). Increasing n from 25 to 100 divides the standard deviation by sqrt(4), so the n = 100 standard deviation is half as large.

Effect of sample size on standard deviation of sample proportions: in the "Suppose the true proportion of residents in a town who recycle batteries is p. Two sampling plans take independent random samples, one with n = 25 residents and one with n = 100 residents. How does the standard deviation of the sample" setting, especially sample proportion; residents recycling batteries, identify the sampling-distribution center, spread, and conditions before using a model. The professional check is to keep the values 25, 100, the sampling distribution spread method, and the final answer choice aligned with Sampling Distributions.

Common mistakes. For the stem cue "Suppose the true proportion of residents in a town who recycle batteries is p. Two sampling plans take independent random samples, one with n = twenty five residents and one with n", a likely mistake is the variance is one-fourth as large, but the standard deviation is the square root of that. Keep the calculation or conclusion tied to the original context.

70 questions on this unit, free to start.

Ten a day at no cost. One account covers SAT, ACT and AP.

Start free
FAQ

Sampling Distributions questions

What is a sampling distribution?

The distribution of a statistic — a sample mean or proportion — across all possible samples of a given size, not the distribution of the data itself.

What does the Central Limit Theorem say?

That the sampling distribution of the sample mean approaches normal as sample size grows, whatever the population’s shape.
Last reviewed 2026-08-28. Unit and topic names follow the College Board course framework. Question counts describe the PrepScore practice bank, not the exam.
Topic by topic

Worked questions by topic

1 of this unit’s topics have enough practice questions for a page of their own.

Sampling Distributions for Sample Proportions49 questions
Same course
Unit 1: Exploring One-Variable Data460 questionsUnit 2: Exploring Two-Variable Data295 questionsUnit 3: Collecting Data190 questionsUnit 4: Probability, Random Variables, and Probability Distributions100 questionsUnit 6: Inference for Categorical Data: Proportions107 questionsUnit 7: Inference for Quantitative Data: Means63 questionsUnit 9: Inference for Quantitative Data: Slopes133 questions

Practise sampling distributions until the reasoning is automatic.

Real AP questions with a full explanation on every answer, and a mistake bank that only clears when you get it right.

Start freeSee pricing