AP Statistics Practice Quiz: The Central Limit Theorem

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 12 questions to check your progress.

Question 1 of 12

Which of the following best defines the 'sampling distribution of a statistic'?

A)The distribution of values within one specific sample of a given size.B)The distribution of values for a statistic from all possible samples of a given size.C)A statement that the distribution of sample means will be approximately normal.D)A collection of statistics generated by assuming a known parameter value.

All Questions (12)

Which of the following best defines the 'sampling distribution of a statistic'?

A) The distribution of values within one specific sample of a given size.

B) The distribution of values for a statistic from all possible samples of a given size.

C) A statement that the distribution of sample means will be approximately normal.

D) A collection of statistics generated by assuming a known parameter value.

Correct Answer: B

This is the direct definition provided by the content, which states that a sampling distribution of a statistic is 'the distribution of values for that statistic from all possible samples of a given size.'

The Central Limit Theorem (CLT) primarily describes the shape of which distribution when the sample size is large?

A) The population distribution.

B) The distribution of a single sample.

C) The sampling distribution of the mean.

D) The randomization distribution.

Correct Answer: C

The content explicitly states that the Central Limit Theorem (CLT) 'states that for a large enough sample size, the sampling distribution of the mean will be approximately normal.'

A researcher wants to understand the distribution of the sample proportion. How could they simulate or estimate the sampling distribution of the proportion?

A) By taking one single, very large random sample from the population.

B) By generating repeated random samples from the population and calculating the proportion for each.

C) By ensuring the sample values are independent and the sample size is large.

D) By creating a distribution assuming the population proportion is a specific, known value.

Correct Answer: B

The content specifies that 'the sampling distribution of a statistic can be simulated by generating repeated random samples from a population.' Calculating the statistic (in this case, the proportion) for each of those samples creates the simulated distribution.

A population's distribution is known to be heavily skewed, not normal. If a researcher takes a very large random sample from this population, what does the Central Limit Theorem suggest about the shape of the sampling distribution of the sample mean?

A) It will also be heavily skewed, just like the population.

B) It will be approximately normal.

C) It will be a uniform distribution.

D) Its shape cannot be determined from the information given.

Correct Answer: B

According to the Central Limit Theorem, 'for a large enough sample size, the sampling distribution of the mean will be approximately normal.' This holds true even if the original population is not normal.

For the Central Limit Theorem to be applicable, which two conditions are required?

A) A normal population and a small sample size.

B) A known population parameter and dependent sample values.

C) Independent sample values and a sufficiently large sample size.

D) A simulated distribution and a known statistic.

Correct Answer: C

The provided content clearly states that 'The CLT requires independent sample values and a sufficiently large sample size (n).'

A statistician is testing a hypothesis that a coin is fair (p=0.5). They simulate flipping the coin 100 times, record the proportion of heads, and repeat this process 10,000 times. What is this collection of 10,000 proportions called?

A) A sampling distribution.

B) A randomization distribution.

C) A population distribution.

D) A sample distribution.

Correct Answer: B

The content defines a randomization distribution as 'a collection of statistics generated by simulation assuming known parameter values.' Here, the simulation assumes the known parameter p=0.5.

A researcher collects a random sample of 50 fish from a lake and calculates their average length to be 10.2 inches. This value of 10.2 inches represents one data point from which of the following?

A) The sampling distribution of the mean length.

B) The population distribution of all fish lengths.

C) A randomization distribution of fish lengths.

D) The distribution of all possible sample sizes.

Correct Answer: A

The average length (10.2 inches) is a statistic calculated from a single sample. The sampling distribution is the distribution of this statistic from all possible samples, so the single calculated value is one point within that larger distribution.

A researcher wants to estimate the average salary at a company. They survey a group of 40 employees who are all from the same department and work on the same team. Why might the Central Limit Theorem not be applicable for making inferences from this sample?

A) The sample size of 40 is not sufficiently large.

B) The sample values may not be independent.

C) The population of salaries is likely skewed.

D) The sampling distribution was not generated using simulation.

Correct Answer: B

The CLT requires independent sample values. Since the employees are all from the same team and department, their salaries may be related (e.g., similar roles, same manager), violating the independence condition.

If a researcher uses a computer to repeatedly draw random samples of size n=200 from a population and plots the means of these samples, the resulting plot is an estimate of the sampling distribution. According to the Central Limit Theorem, what shape will this plot most likely have?

A) The same shape as the population distribution.

B) A bimodal distribution.

C) A uniform distribution.

D) An approximately normal distribution.

Correct Answer: D

This question combines two concepts. The process described is how to 'estimate sampling distributions using simulation.' The CLT predicts the result of this process for the mean: with a large sample size (n=200), 'the sampling distribution of the mean will be approximately normal.'

A student generates 1,000 random samples of size 30 from a dataset and calculates the standard deviation for each of the 1,000 samples. The collection of these 1,000 standard deviation values represents an approximation of what?

A) The Central Limit Theorem.

B) The sampling distribution of the standard deviation.

C) A randomization distribution of the mean.

D) The distribution of the original population.

Correct Answer: B

The process involves generating repeated random samples and calculating a statistic (the standard deviation) for each. The resulting collection of statistics is an estimate of the sampling distribution of that specific statistic, as described in the content.

Which statement accurately distinguishes between a sampling distribution and a randomization distribution as described in the provided text?

A) A sampling distribution is theoretical, while a randomization distribution is always generated by simulation.

B) A sampling distribution is based on repeated sampling from a population, while a randomization distribution is based on simulation assuming a specific parameter value is true.

C) A sampling distribution is always normal, while a randomization distribution is always skewed.

D) A sampling distribution is used for means, while a randomization distribution is used for proportions.

Correct Answer: B

The content defines a sampling distribution based on 'all possible samples' from a population. It defines a randomization distribution as being 'generated by simulation assuming known parameter values.' This assumption of a known parameter is the key difference.

The Central Limit Theorem is considered powerful for statistical inference because it allows researchers to model the behavior of sample means even if the original population is not normally distributed. What is the primary justification for this?

A) As long as the sample size is large enough, the sampling distribution of the mean will be approximately normal.

B) Any large sample will have a distribution that is approximately normal, regardless of the population shape.

C) All randomization distributions are approximately normal, which allows for accurate inference.

D) The requirement of independent sample values forces the sampling distribution to be normal.

Correct Answer: A

The core utility of the CLT is that the normality of the sampling distribution of the mean depends on the sample size, not the shape of the population. The content states that for a large n, this distribution 'will be approximately normal,' which is the foundation for many inferential procedures.