AP Statistics Practice Quiz: Sampling Distributions for Sample Means

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

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

A population has a mean (μ) of 100 and a standard deviation (σ) of 15. If a random sample of size n = 25 is taken from this population, what is the mean of the sampling distribution of the sample mean (x-bar)?

A)4B)15C)25D)100

All Questions (16)

A) 4

B) 15

C) 25

D) 100

Correct Answer: D

According to the provided content, the sampling distribution of the sample mean (x-bar) has a mean equal to the population mean (μ). In this case, μ = 100, so the mean of the sampling distribution is 100.

The weights of a certain type of apple are normally distributed with a mean of 8 ounces and a standard deviation of 1 ounce. If random samples of 16 apples are selected, what is the standard deviation of the sampling distribution of the sample mean weight?

A) 0.25 ounces

B) 0.5 ounces

C) 1 ounce

D) 4 ounces

Correct Answer: A

The provided content states that the standard deviation of the sampling distribution of the sample mean is σ/√n. Here, the population standard deviation (σ) is 1 ounce and the sample size (n) is 16. Therefore, the standard deviation of the sampling distribution is 1/√16 = 1/4 = 0.25 ounces.

The distribution of annual incomes for adults in a certain city is strongly skewed to the right with a mean of $52,000 and a standard deviation of $18,000. A random sample of 100 adults is taken. Which of the following best describes the sampling distribution of the sample mean income?

A) Strongly skewed to the right with a mean of $52,000 and a standard deviation of $1,800.

B) Approximately normal with a mean of $52,000 and a standard deviation of $18,000.

C) Approximately normal with a mean of $52,000 and a standard deviation of $1,800.

D) Shape is unknown, but the mean is $52,000 and the standard deviation is $1,800.

Correct Answer: C

The content states that if the population distribution is not normal, the sampling distribution of the sample mean is approximately normal if the sample size is large enough (n ≥ 30). Since n = 100, the condition is met. The mean of the sampling distribution is μ = $52,000, and the standard deviation is σ/√n = $18,000/√100 = $1,800.

A population's distribution is known to be normal. A random sample of size n = 10 is drawn from this population. What is the shape of the sampling distribution of the sample mean?

A) Normal

B) Approximately normal, but not exactly normal because the sample size is small.

C) Skewed, because the sample size is less than 30.

D) The shape cannot be determined with a sample size this small.

Correct Answer: A

According to the provided content, if the population distribution is normal, the sampling distribution of the sample mean is also normal, regardless of the sample size. The condition of n ≥ 30 is only necessary when the population distribution is not normal.

A high school has 200 seniors. A researcher takes a random sample of 50 seniors to estimate the mean GPA. The population standard deviation of GPAs is known. Under which condition is the use of the standard deviation formula σ/√n for the sampling distribution potentially inappropriate?

A) The population distribution of GPAs is not normal.

B) The sample was not selected randomly.

C) The sample size is greater than 10% of the population size.

D) The population standard deviation is unknown.

Correct Answer: C

The content states that if sampling without replacement, the standard deviation formula σ/√n is approximately correct if the sample size is less than 10% of the population. Here, the sample size is 50 and the population is 200. The sample size is 50/200 = 25% of the population, which violates the 10% condition, making the formula for standard deviation potentially inaccurate.

The mean score on a standardized test is 1500 with a standard deviation of 200. A random sample of 40 test-takers is selected. The probability that the sample mean score is greater than 1550 is calculated. How should this probability be interpreted?

A) The probability that a single randomly selected test-taker scores above 1550.

B) The probability of obtaining a random sample of 40 test-takers whose average score is greater than 1550.

C) The probability that the true population mean score is actually greater than 1550.

D) The probability that every one of the 40 test-takers in the sample scores above 1550.

Correct Answer: B

The content specifies that probabilities for a sampling distribution for a sample mean should be interpreted in context. The calculation is based on the distribution of sample means (x-bar), not individual scores. Therefore, the probability refers to the likelihood of observing a sample mean within a certain range.

Under which of the following conditions can the sampling distribution of the sample mean NOT be described as approximately normal?

A) The population distribution is normal and the sample size is 15.

B) The population distribution is skewed and the sample size is 45.

C) The population distribution is unknown and the sample size is 50.

D) The population distribution is skewed and the sample size is 20.

Correct Answer: D

The content states that if the population is not normal, the sample size must be large enough (e.g., n ≥ 30) for the sampling distribution of the sample mean to be approximately normal. In option D, the population is skewed (not normal) and the sample size (n=20) is less than 30, so the normality condition is not met.

A researcher is studying the average height of a certain species of plant. The population mean height is 30 cm. The researcher takes many random samples of size n=35 and calculates the mean for each sample. What does the mean of the distribution of all these sample means represent?

A) The mean of the largest sample taken.

B) The standard deviation of the plant heights.

C) The true population mean height of the plants.

D) The sample size.

Correct Answer: C

Based on the provided content, the mean of the sampling distribution of the sample mean is equal to the population mean (μ). Therefore, the mean of all possible sample means is an unbiased estimator of the true population mean, which is 30 cm in this context.

Let X be a random variable representing the battery life of a smartphone, which follows a non-normal distribution with μ = 20 hours and σ = 4 hours. Two researchers take samples. Researcher A takes a sample of size n=15. Researcher B takes a sample of size n=40. Which statement correctly compares the sampling distributions of their sample means?

A) Both distributions will be approximately normal.

B) The mean of A's distribution will be smaller than the mean of B's distribution.

C) The standard deviation of A's distribution will be larger than the standard deviation of B's distribution.

D) The distribution for Researcher A will be approximately normal, but the distribution for B will not.

Correct Answer: C

The mean of both sampling distributions will be μ = 20. The standard deviation is σ/√n. For A, it's 4/√15 ≈ 1.03. For B, it's 4/√40 ≈ 0.63. Thus, the standard deviation for A's smaller sample is larger. Additionally, only B's distribution will be approximately normal because its sample size (n=40) is ≥ 30, while A's (n=15) is not.

The time it takes for a delivery service to deliver a package is normally distributed with a mean of 48 hours and a standard deviation of 6 hours. A random sample of 9 packages is selected. What are the parameters (mean and standard deviation) of the sampling distribution of the sample mean delivery time?

A) mean = 48, standard deviation = 6

B) mean = 48, standard deviation = 2

C) mean = 16, standard deviation = 6

D) mean = 16, standard deviation = 2

Correct Answer: B

The content states that the mean of the sampling distribution is equal to the population mean (μ), and the standard deviation is σ/√n. Here, μ = 48 and σ = 6. With a sample size n = 9, the mean of the sampling distribution is 48 and the standard deviation is 6/√9 = 6/3 = 2.

The standard deviation of the sampling distribution of the sample mean is often called the standard error of the mean. What does this value measure in context?

A) The variability of individual data points within the population.

B) The typical distance between a sample mean (x-bar) and the population mean (μ).

C) The error made in calculating the population mean.

D) The degree to which the population distribution deviates from normal.

Correct Answer: B

The standard deviation of a sampling distribution (σ/√n) measures the variability of the statistic (in this case, x-bar) across all possible samples of a given size. Interpreted in context, it represents the typical or average distance of the sample means from the true population mean.

A student wants to determine if the sampling distribution of the sample mean can be considered approximately normal. The population distribution is unknown. Which of the following sample sizes is most likely large enough to satisfy the condition for approximate normality?

A) n = 5

B) n = 15

C) n = 25

D) n = 35

Correct Answer: D

According to the content, if the population distribution is not normal (or unknown), the sampling distribution of the sample mean is approximately normal if the sample size is large enough. The guideline provided is n ≥ 30. Of the options given, only n = 35 meets this condition.

If all other factors remain constant, how does increasing the sample size from n=25 to n=100 affect the standard deviation of the sampling distribution of the sample mean?

A) It will be multiplied by 4.

B) It will be multiplied by 2.

C) It will be divided by 2.

D) It will be divided by 4.

Correct Answer: C

The standard deviation of the sampling distribution is σ/√n. The original standard deviation is σ/√25 = σ/5. The new standard deviation is σ/√100 = σ/10. To get from σ/5 to σ/10, you divide by 2. Therefore, increasing the sample size by a factor of 4 decreases the standard deviation by a factor of √4 = 2.

A factory produces widgets with a mean weight of 50 grams and a standard deviation of 5 grams. The distribution of weights is not normal. An inspector plans to take a random sample of 20 widgets. He calculates the probability that the sample mean is less than 49 grams using a normal distribution. Why is this calculation potentially invalid?

A) The population mean is not large enough.

B) The population standard deviation is too large relative to the mean.

C) The sample size is not large enough to assume the sampling distribution is approximately normal.

D) The sample was taken without replacement.

Correct Answer: C

The content states that if the population distribution is not normal, the sampling distribution of the sample mean is approximately normal only if the sample size is large enough (e.g., n ≥ 30). Since the population is not normal and the sample size is only 20, the Central Limit Theorem does not apply, and using a normal distribution for probability calculations is not justified.

Which of the following statements correctly describes a parameter of the sampling distribution of the sample mean, x-bar?

A) The mean of the sampling distribution is μ/√n.

B) The standard deviation of the sampling distribution is σ.

C) The mean of the sampling distribution is equal to the population mean, μ.

D) The shape of the sampling distribution is always the same as the population shape.

Correct Answer: C

The provided content explicitly states that the sampling distribution of the sample mean x-bar has a mean equal to μ (the population mean) and a standard deviation of σ/√n. Option C correctly identifies the mean of the sampling distribution.

A small college has 150 employees. The mean salary is $60,000 with a standard deviation of $10,000. A random sample of 30 employees is selected to estimate the mean salary. A statistician notes that the 10% condition has been violated. What is the primary consequence of this violation?

A) The mean of the sampling distribution will no longer be $60,000.

B) The shape of the sampling distribution cannot be assumed to be approximately normal.

C) The calculated standard deviation of the sampling distribution, σ/√n, may not be an accurate measure of the variability of the sample mean.

D) The sample mean will be a biased estimator of the population mean.

Correct Answer: C

The content states that the formula σ/√n for the standard deviation is approximately correct if the sample size is less than 10% of the population when sampling without replacement. Here, n=30 is 20% of the population of 150. Violating this condition means the sample selections are not independent, and the formula σ/√n will overestimate the true standard deviation of the sampling distribution.