AP Statistics Practice Quiz: Sampling Distributions for Differences in 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 researcher is comparing the mean heights of adult males from two different countries. The mean height in Country 1 is μ₁ = 175 cm, and the mean height in Country 2 is μ₂ = 170 cm. If random samples are taken from both populations, what is the mean of the sampling distribution of the difference in sample means (x̄₁ - x̄₂)?

A)5 cmB)172.5 cmC)345 cmD)It cannot be determined without the sample sizes.

All Questions (16)

A) 5 cm

B) 172.5 cm

C) 345 cm

D) It cannot be determined without the sample sizes.

Correct Answer: A

The mean of the sampling distribution of the difference in sample means (x̄₁ - x̄₂) is equal to the difference in the population means (μ₁ - μ₂). In this case, it is 175 cm - 170 cm = 5 cm.

The distribution of weights for a species of bird in Region A is normally distributed. The distribution of weights for the same species in Region B is also normally distributed. A researcher takes a sample of 10 birds from Region A and 12 birds from Region B. What is the shape of the sampling distribution of the difference in sample mean weights (x̄ₐ - x̄ₑ)?

A) Approximately normal because the sample sizes are large enough.

B) Normal because both population distributions are normal.

C) Skewed, because the sample sizes are small.

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

Correct Answer: B

If both population distributions are normal, the sampling distribution of the difference in means is also normal, regardless of the sample sizes.

A study compares the mean battery life of two brands of smartphones. The distribution of battery life for both brands is known to be skewed right. A researcher takes a random sample of 40 phones from Brand X and 50 phones from Brand Y. Which of the following statements best describes the sampling distribution of the difference in sample means (x̄ₓ - x̄ᵧ)?

A) It is skewed right because both population distributions are skewed right.

B) It is approximately normal because both sample sizes are large enough.

C) It is normal because the samples are random.

D) Its shape cannot be determined because the population distributions are not normal.

Correct Answer: B

According to the Central Limit Theorem, if the population distributions are not normal, the sampling distribution of the difference in means is approximately normal as long as both sample sizes are large enough (typically n ≥ 30). Here, n₁ = 40 and n₂ = 50, so the condition is met.

A high school has 1,200 senior students and 1,300 junior students. To compare the mean GPA of the two classes, a counselor selects a random sample of 50 seniors and 50 juniors without replacement. To use the standard deviation formula for the difference in sample means, which condition must be acknowledged?

A) The sample sizes must be greater than or equal to 30.

B) The population distributions of GPA for seniors and juniors must be normal.

C) The sample sizes should be less than 10% of their respective population sizes.

D) The total number of students sampled must be less than 10% of the total school population.

Correct Answer: C

When sampling without replacement, the formula for the standard deviation of the sampling distribution is approximately correct if the sample sizes are less than 10% of their population sizes. This is known as the 10% condition, which helps ensure the independence of the observations. 50 is less than 10% of 1,200 and less than 10% of 1,300.

The true mean daily screen time for teenagers is 7.5 hours, and for adults is 4.5 hours. For random samples of 100 teenagers and 100 adults, the probability of observing a difference in sample means (teenager - adult) of 4 hours or more is calculated to be 0.04. Which of the following is the correct interpretation of this probability?

A) There is a 4% chance that the true difference in mean screen time between all teenagers and all adults is 4 hours or more.

B) If we take many random samples of 100 teenagers and 100 adults, about 4% of those samples will result in a difference in sample means of 4 hours or more.

C) There is a 4% chance that a randomly selected teenager will have at least 4 hours more screen time than a randomly selected adult.

D) In 4% of all possible samples, the sample mean for teenagers will be exactly 4 hours greater than the sample mean for adults.

Correct Answer: B

A probability from a sampling distribution refers to the likelihood of observing a particular sample statistic (or one more extreme) in repeated sampling. It describes the behavior of the sample difference, not the true population difference or the difference between individuals.

For two independent populations, the mean of Population 1 is μ₁ and the mean of Population 2 is μ₂. Which of the following represents the mean of the sampling distribution of the difference in sample means, x̄₁ - x̄₂?

A) μ₁ + μ₂

B) μ₁ - μ₂

C) (μ₁ + μ₂) / 2

D) It depends on the sample sizes n₁ and n₂.

Correct Answer: B

The mean of the sampling distribution for the difference in sample means is a direct reflection of the difference between the two population means. Therefore, its value is μ₁ - μ₂.

A researcher wants to compare the mean scores on a memory test for two groups. Group A's scores are from a population with a skewed-left distribution, while Group B's scores are from a population with a skewed-right distribution. The researcher takes a sample of 20 from Group A and 25 from Group B. Why is it inappropriate to assume the sampling distribution of the difference in sample means is approximately normal?

A) Because the population distributions have different shapes.

B) Because the sample sizes are not equal.

C) Because the samples were not taken from normal populations and the sample sizes are not large enough.

D) Because the 10% condition for independence was not checked.

Correct Answer: C

The Central Limit Theorem (CLT) allows for an approximately normal sampling distribution if sample sizes are large (e.g., ≥ 30), even if the underlying populations are not normal. Since both sample sizes (20 and 25) are less than 30 and the populations are not normal, the CLT does not apply.

The mean salary for statisticians in State A is $80,000, and in State B is $85,000. The population distributions of salaries in both states are known to be right-skewed. If a researcher takes a random sample of 100 statisticians from each state, which of the following statements about the sampling distribution of the difference in sample means (x̄ₐ - x̄ₑ) is most accurate?

A) The distribution will be right-skewed with a mean of -$5,000.

B) The distribution will be approximately normal with a mean of -$5,000.

C) The distribution will be approximately normal with a mean of $5,000.

D) The shape of the distribution cannot be determined, but the mean is -$5,000.

Correct Answer: B

The mean of the sampling distribution is μₐ - μₑ = $80,000 - $85,000 = -$5,000. Because both sample sizes (n=100) are large (≥ 30), the Central Limit Theorem applies, and the sampling distribution of the difference in means will be approximately normal, despite the skewed population distributions.

Suppose the mean score on a standardized test for students who used a new curriculum is μ₁ = 520, and for students who used the old curriculum is μ₂ = 500. This means the center of the sampling distribution for the difference in sample means (x̄₁ - x̄₂) is 20. What is the correct interpretation of this parameter?

A) Every sample of students from the new curriculum will have a mean score 20 points higher than every sample from the old curriculum.

B) The most likely difference to be observed between any two individual students from each group is 20 points.

C) Over many pairs of random samples, the average difference between the sample means will be 20 points.

D) The standard deviation of the difference in sample means is 20 points.

Correct Answer: C

The mean of a sampling distribution represents the long-run average of the sample statistic over many repeated samples. Therefore, a mean of 20 implies that, on average, the difference between the sample means (x̄₁ - x̄₂) will be 20 points.

A researcher is comparing the mean number of defective products from two different assembly lines, A and B. Both production lines produce thousands of items per day. A sample of 100 items is taken from Line A, and 150 items are taken from Line B. Both population distributions of defects are non-normal. To calculate a probability about the difference in sample means, which set of conditions is required?

A) Both populations must be normal, and samples must be random.

B) The sample sizes must be large, and the samples must be random.

C) The samples must be random, sample sizes must be large, and the 10% condition must be met for calculating standard deviation.

D) The sample sizes must be equal, and the samples must be random.

Correct Answer: C

To model the sampling distribution, three conditions are key: (1) Random samples to ensure they are representative. (2) The Normality/Large Counts condition (here, large sample sizes of 100 and 150 satisfy the CLT since populations are non-normal). (3) The 10% condition (n < 0.1N) to ensure independence of observations when sampling without replacement, which is necessary for the standard deviation formula.

The sampling distribution of the difference in means is guaranteed to be normal if which of the following is true?

A) Both sample sizes are greater than or equal to 30.

B) The samples are selected randomly.

C) Both population distributions are normal.

D) The sample sizes are less than 10% of the population sizes.

Correct Answer: C

While large sample sizes (n ≥ 30) make the sampling distribution *approximately* normal due to the CLT, it is *exactly* normal if the two underlying population distributions are themselves normal. This holds true for any sample size.

The mean weight of apples from Orchard A is 150g, and from Orchard B is 140g. For random samples of 50 apples from each orchard, the standard deviation of the sampling distribution of the difference in sample means (x̄ₐ - x̄ₑ) is 3g. Which of the following is the best interpretation of this standard deviation?

A) The difference between the weights of any single apple from Orchard A and any single apple from Orchard B is typically 3g.

B) The standard deviation of apple weights within Orchard A differs from Orchard B by 3g.

C) In repeated random samples of size 50 from each orchard, the difference in sample means typically varies from the true difference in population means (10g) by about 3g.

D) The average weight of apples in a sample of 50 from either orchard will typically be 3g away from the population mean.

Correct Answer: C

The standard deviation of a sampling distribution measures the typical or average distance between the sample statistic (in this case, the difference in sample means) and the population parameter (the true difference in population means). It describes the variability of the sample statistic, not of individual data points.

A researcher samples 400 voters from a population of 5 million Democrats and 400 voters from a population of 4 million Republicans. The samples are taken without replacement. Why is the use of the standard deviation formula for the difference in sample means justified in this context?

A) Because the sample sizes are large (n ≥ 30).

B) Because the samples are random.

C) Because the sample sizes are less than 10% of their respective population sizes.

D) Because the population sizes are not equal.

Correct Answer: C

The standard deviation formula for the difference in sample means assumes that the observations are independent. When sampling without replacement, this independence is violated, but the formula is still approximately correct if the sample size is no more than 10% of the population size. Here, 400 is much less than 10% of 5 million and 4 million, so the condition is met.

The distribution of daily rainfall in City 1 is normal with a mean of 0.2 inches. The distribution of daily rainfall in City 2 is also normal with a mean of 0.15 inches. Random samples of 20 days are taken from each city. Which statement correctly describes the parameters of the sampling distribution of the difference in sample means (x̄₁ - x̄₂)?

A) The mean is 0.05 inches, and the shape is approximately normal.

B) The mean is 0.05 inches, and the shape is normal.

C) The mean is 0.35 inches, and the shape is normal.

D) The mean is 0.05 inches, but the shape cannot be determined because the sample sizes are small.

Correct Answer: B

The mean of the sampling distribution is the difference in population means: μ₁ - μ₂ = 0.2 - 0.15 = 0.05 inches. Because both population distributions are stated to be normal, the sampling distribution of the difference in means is also normal, regardless of the sample sizes.

A student is comparing the average cost of a one-bedroom apartment in two different cities. The distribution of rent is heavily right-skewed in both cities. The student takes a random sample of 15 apartments from the first city and 20 from the second. The student wants to construct a confidence interval for the difference in mean rent, which relies on the sampling distribution. What is the primary reason the student should be hesitant to proceed?

A) The samples were random, which introduces too much variability.

B) The sample sizes are too small to overcome the skewness of the populations, so the sampling distribution may not be approximately normal.

C) The 10% condition was not checked, so the standard deviation cannot be calculated.

D) The sample sizes are unequal, which is not allowed when comparing two means.

Correct Answer: B

The validity of many inference procedures for means, including confidence intervals and hypothesis tests, depends on the sampling distribution being at least approximately normal. Since the populations are heavily skewed and the sample sizes (15 and 20) are not large enough (i.e., not ≥ 30), the Central Limit Theorem does not apply, and the normality condition is not met.

Let μ₁ be the true mean reaction time for a task using the left hand and μ₂ be the true mean reaction time using the right hand. A researcher finds that the mean of the sampling distribution for the difference in sample means (x̄₁ - x̄₂) is 0.02 seconds. Which of the following is a correct statement based on this parameter?

A) The true mean reaction time for the left hand is 0.02 seconds.

B) The true difference in mean reaction times between the left and right hands is 0.02 seconds.

C) For any pair of samples, the difference in sample means will be 0.02 seconds.

D) The probability of observing a difference of exactly 0.02 seconds is very high.

Correct Answer: B

The mean of the sampling distribution of the difference in sample means is equal to the difference in the population means. Therefore, if the mean of the sampling distribution of (x̄₁ - x̄₂) is 0.02, then the true difference in the population parameters (μ₁ - μ₂) must also be 0.02.