AP Statistics Practice Quiz: Sampling Distributions for Sample Proportions

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 large high school reports that 85% of its students have a smartphone. A random sample of 50 students is taken. What is the mean of the sampling distribution of the sample proportion of students who have a smartphone?

A)0.85B)42.5C)0.048D)It cannot be determined without the sample data.

All Questions (16)

A) 0.85

B) 42.5

C) 0.048

D) It cannot be determined without the sample data.

Correct Answer: A

According to the provided content, the mean of the sampling distribution of a sample proportion (p-hat) is equal to the population proportion (p). In this case, p = 0.85.

A national survey indicates that 30% of adults in a country own a tablet. For random samples of size 100, what is the standard deviation of the sampling distribution of the sample proportion of adults who own a tablet?

A) sqrt(0.30(0.70)/100)

B) sqrt(100(0.30)(0.70))

C) 0.30

D) (0.30)(0.70)/100

Correct Answer: A

The formula for the standard deviation of the sampling distribution of a sample proportion p-hat is sqrt(p(1-p)/n). Here, p = 0.30 and n = 100, so the standard deviation is sqrt(0.30(1-0.30)/100).

A manufacturer claims that only 4% of its microchips are defective. A quality control specialist takes a random sample of 200 microchips. Can the sampling distribution of the sample proportion of defective microchips be described as approximately normal?

A) Yes, because the sample size of 200 is large.

B) Yes, because n(1-p) is greater than 10.

C) No, because np is less than 10.

D) No, because the sample is less than 10% of the population.

Correct Answer: C

To determine if the sampling distribution of p-hat is approximately normal, we must check the Large Counts Condition: np >= 10 and n(1-p) >= 10. In this case, n=200 and p=0.04. So, np = 200(0.04) = 8. Since 8 is not greater than or equal to 10, the condition is not met, and the distribution cannot be described as approximately normal.

A university has 20,000 undergraduate students. A researcher takes a random sample of 500 students. When calculating the standard deviation of the sampling distribution for a sample proportion, the researcher confirms that the 10% condition is met. What is the justification for checking this condition?

A) It ensures that the sampling distribution is approximately normal.

B) It ensures that the mean of the sampling distribution is equal to the population proportion.

C) It ensures that the standard deviation formula is approximately correct when sampling without replacement.

D) It ensures that the sample proportion is an unbiased estimator of the population proportion.

Correct Answer: C

The content states that if sampling without replacement and the sample size is less than 10% of the population, the standard deviation formula, sqrt(p(1-p)/n), is approximately correct. This is the purpose of the 10% condition.

The proportion of registered voters in a large district who are Democrats is 0.55. A random sample of 250 voters is selected. Which of the following best describes the sampling distribution of the sample proportion of Democrats?

A) Approximately normal with mean 0.55 and standard deviation sqrt(0.55(0.45)/250).

B) Approximately normal with mean 137.5 and standard deviation sqrt(250(0.55)(0.45)).

C) Shape is unknown, with mean 0.55 and standard deviation sqrt(0.55(0.45)/250).

D) Exactly normal with mean 0.55 and standard deviation sqrt(0.55(0.45)/250).

Correct Answer: A

The mean of the sampling distribution is p = 0.55. The standard deviation is sqrt(p(1-p)/n) = sqrt(0.55(0.45)/250). To check for normality, we use the Large Counts Condition: np = 250(0.55) = 137.5 >= 10 and n(1-p) = 250(0.45) = 112.5 >= 10. Since both are met, the distribution is approximately normal.

A national polling organization reports that 60% of adults favor a certain public policy. If numerous random samples of 400 adults are taken, the standard deviation of the sampling distribution of the sample proportion is approximately 0.024. Which of the following is the best interpretation of this value?

A) In any given sample of 400 adults, the sample proportion will be 0.024.

B) The largest possible sample proportion is approximately 0.624.

C) The distance between any two sample proportions will be, on average, about 0.024.

D) The typical distance between a sample proportion and the true proportion of 0.60 is about 0.024.

Correct Answer: D

The standard deviation of a sampling distribution measures the typical or average distance of the statistic (in this case, the sample proportion p-hat) from the parameter (the population proportion p). Therefore, it represents the typical distance between a sample proportion and the true proportion.

The Large Counts Condition is used to determine whether a sampling distribution for a sample proportion can be described as approximately normal. What is this condition?

A) n < 0.10N

B) n >= 30

C) np >= 10 and n(1-p) >= 10

D) p-hat must be close to p

Correct Answer: C

The provided content explicitly states that the sampling distribution of p-hat is approximately normal if np >= 10 and n(1-p) >= 10. This is known as the Large Counts Condition.

A small liberal arts college has 800 students. A sociologist takes a random sample of 100 students to estimate the proportion who are first-generation college students. Why is it potentially problematic to use the formula sqrt(p(1-p)/n) to calculate the standard deviation of the sampling distribution?

A) The sample size is not large enough.

B) The population proportion p is unknown.

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

D) The Large Counts Condition is not met.

Correct Answer: C

The 10% condition must be met when sampling without replacement to use the standard deviation formula. Here, the sample size n=100 and the population size N=800. Since 100 is greater than 10% of 800 (which is 80), the condition is not met, and the formula will not be approximately correct.

Suppose for a certain population proportion p and sample size n, the probability of observing a sample proportion (p-hat) of 0.75 or greater is 0.05. Which is the correct interpretation of this probability?

A) There is a 5% chance that the true population proportion is 0.75 or greater.

B) If we take one sample, there is a 5% chance its proportion will be exactly 0.75.

C) In about 5% of all possible random samples of size n, the sample proportion will be 0.75 or greater.

D) The population proportion must be less than 0.75.

Correct Answer: C

Probabilities for a sampling distribution describe what would happen in the long run over many repeated random samples. The value 0.05 represents the long-run relative frequency of observing a sample proportion in the specified range (0.75 or greater) from repeated samples of the same size n.

Which of the following correctly identifies the parameters of the sampling distribution of a sample proportion, p-hat?

A) Mean = p, Standard Deviation = sqrt(p(1-p)/n)

B) Mean = p-hat, Standard Deviation = sqrt(p-hat(1-p-hat)/n)

C) Mean = np, Standard Deviation = sqrt(np(1-p))

D) Mean = p, Standard Deviation = p(1-p)/n

Correct Answer: A

The content specifies that the sampling distribution of a sample proportion p-hat has a mean equal to the population proportion, p, and a standard deviation equal to sqrt(p(1-p)/n).

A seed company claims that 92% of its seeds will germinate. A gardener plants 150 seeds. Assuming the company's claim is true, which of the following statements about the sampling distribution of the proportion of seeds that germinate is correct?

A) The distribution is not approximately normal because p is too close to 1.

B) The mean is 0.92, but the shape of the distribution is skewed to the left.

C) The distribution is approximately normal with a mean of 0.92.

D) The standard deviation cannot be calculated because the 10% condition cannot be checked.

Correct Answer: C

The mean of the sampling distribution is p = 0.92. To check for normality, we use the Large Counts Condition: np = 150(0.92) = 138 >= 10, and n(1-p) = 150(0.08) = 12 >= 10. Since both conditions are met, the sampling distribution can be described as approximately normal.

A researcher wants to estimate the proportion of fish in a large lake with a certain genetic marker, believed to be present in about 8% of the fish. What is the minimum sample size the researcher should take to ensure the sampling distribution of the sample proportion can be approximated by a normal model?

A) 30

B) 100

C) 125

D) 200

Correct Answer: C

To use a normal model, the Large Counts Condition must be met: np >= 10 and n(1-p) >= 10. Since p=0.08 is the smaller value, the limiting condition is np >= 10. We solve for n: n(0.08) >= 10, which gives n >= 10/0.08, or n >= 125. Therefore, the minimum sample size is 125.

In the context of a sampling distribution for a sample proportion, what do p and p-hat represent?

A) p is the sample proportion and p-hat is the population proportion.

B) p is the population proportion and p-hat is the sample proportion.

C) Both p and p-hat are sample proportions from different samples.

D) Both p and p-hat are parameters of the population.

Correct Answer: B

By definition, p is the parameter that represents the true proportion in the population. The statistic p-hat is the sample proportion, which is calculated from a sample and is used to estimate p. The sampling distribution describes the behavior of the statistic p-hat.

It is believed that 20% of all emails are spam. A company's server processes 500 emails. If we consider this a random sample, what is the interpretation of the mean of the sampling distribution of the sample proportion of spam emails?

A) The mean number of spam emails in a sample of 500 is 100.

B) If we took many random samples of 500 emails, the average of the sample proportions of spam emails would be 0.20.

C) In a specific sample of 500 emails, we expect exactly 20% to be spam.

D) The mean is 0.20, which is the most likely proportion to occur in any single sample.

Correct Answer: B

The mean of the sampling distribution (μ_p̂ = p) represents the long-run average of the sample statistic (p-hat) over many repeated samples. Therefore, the average of all possible sample proportions would be equal to the true population proportion, 0.20.

Under what condition is the sampling distribution of p-hat considered to be an unbiased estimator of p?

A) When np >= 10 and n(1-p) >= 10.

B) When the sample size is less than 10% of the population.

C) When the data is collected from a simple random sample.

D) When the standard deviation is small.

Correct Answer: C

An estimator is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. The mean of the sampling distribution of p-hat is equal to p (μ_p̂ = p) when the data comes from a random sample. The other conditions relate to the shape (normality) and spread (standard deviation calculation) of the distribution, not its center.

A state's department of motor vehicles reports that 15% of all licensed drivers are teenagers. An insurance company takes a random sample of 300 of its policyholders who are licensed drivers. Which of the following is NOT a correct statement about the sampling distribution of the sample proportion of teenagers?

A) The mean of the sampling distribution is 0.15.

B) The standard deviation of the sampling distribution is sqrt(0.15(0.85)/300).

C) The sampling distribution can be approximated by a Normal model.

D) The shape of the sampling distribution is exactly normal.

Correct Answer: D

The mean is p=0.15. The standard deviation is sqrt(p(1-p)/n). The Large Counts Condition is met (np = 300*0.15 = 45 >= 10 and n(1-p) = 300*0.85 = 255 >= 10), so the distribution is approximately normal. However, sampling distributions of proportions are based on binomial distributions and are only *approximately* normal, not *exactly* normal, regardless of sample size.