AP Statistics Practice Quiz: Confidence Intervals for the Difference of Two 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 wants to estimate the difference in the mean SAT scores between students from two different high schools. The data consists of two independent random samples of students. Which of the following is the most appropriate statistical procedure to use?

A)A one-sample t-interval for a mean.B)A two-sample z-interval for a difference between proportions.C)A two-sample t-interval for a difference between means.D)A chi-square test for independence.

All Questions (16)

A) A one-sample t-interval for a mean.

B) A two-sample z-interval for a difference between proportions.

C) A two-sample t-interval for a difference between means.

D) A chi-square test for independence.

Correct Answer: C

According to the provided content, the appropriate procedure for estimating the difference between two population means from two independent samples is a two-sample t-interval for a difference between means.

When constructing a confidence interval for the difference of two population means, what value serves as the point estimate?

A) The average of the two sample means, (x1-bar + x2-bar) / 2.

B) The difference between the two sample means, x1-bar - x2-bar.

C) The pooled sample standard deviation.

D) The t* critical value.

Correct Answer: B

The content explicitly states that the point estimate for the difference of two population means is x1-bar - x2-bar.

Which of the following are the primary conditions that must be verified before calculating a confidence interval for a difference of two population means?

A) The samples must be dependent, and the populations must be skewed.

B) The population standard deviations must be known, and the sample sizes must be small.

C) The groups must be independent, and the sampling distribution of the difference must be approximately normal.

D) The sample means must be equal, and the data must be categorical.

Correct Answer: C

The provided content specifies that to estimate a difference of means, one must check for the independence of groups and that the sampling distribution is approximately normal.

The margin of error for the difference of two means is calculated by multiplying the standard error of the difference by which other value?

A) The sample size (n)

B) The point estimate (x1-bar - x2-bar)

C) The critical value (t*)

D) The significance level (alpha)

Correct Answer: C

The content states that the margin of error for the difference of two means is t* times the standard error of the difference.

A confidence interval for the difference of two means is constructed using the formula: (point estimate) ± (margin of error). Which of the following correctly represents this structure?

A) (x1-bar - x2-bar) ± t* * SE

B) (x1-bar + x2-bar) ± t* * SE

C) t* ± (x1-bar - x2-bar) * SE

D) SE ± t* * (x1-bar - x2-bar)

Correct Answer: A

The content specifies that the confidence interval for a difference of means is calculated as (x1-bar - x2-bar) ± t* * SE.

The formula for the standard error of the difference in two sample means is based on which components from the two samples?

A) The sample medians and interquartile ranges.

B) The sample means and the confidence level.

C) The sample standard deviations and sample sizes.

D) The population means and population standard deviations.

Correct Answer: C

The content mentions that the standard error for the difference in two sample means has a specific formula involving the sample standard deviations. The formula also requires the sample sizes.

When calculating a two-sample t-interval for a difference of means, the degrees of freedom for finding the t* critical value are complex. According to the provided information, how should these degrees of freedom be determined?

A) By subtracting 2 from the sum of the sample sizes (n1 + n2 - 2).

B) By using the smaller of n1 - 1 or n2 - 1.

C) By using technology or a statistical software package.

D) By averaging the two sample sizes and subtracting 1.

Correct Answer: C

The content explicitly states that for a confidence interval for a difference of means, the degrees of freedom are found using technology.

What is the formal name of the confidence interval procedure used to estimate the difference between two population means when dealing with two independent samples?

A) A matched-pairs t-interval.

B) A two-sample t-interval for a difference between means.

C) A two-sample z-interval for a difference between means.

D) A confidence interval for a slope.

Correct Answer: B

The content identifies the appropriate procedure for two independent samples as a two-sample t-interval for a difference between means.

A study compared the mean hours of sleep for a sample of 40 college freshmen and an independent sample of 35 college seniors. The freshmen had a mean of 6.8 hours and the seniors had a mean of 7.1 hours. What is the point estimate for the difference in mean hours of sleep (freshmen - seniors)?

A) 6.95 hours

B) 0.3 hours

C) -0.3 hours

D) 13.9 hours

Correct Answer: C

The point estimate is the difference between the two sample means, x1-bar - x2-bar. In this case, it is 6.8 - 7.1 = -0.3 hours.

A researcher has calculated the components for a 95% confidence interval for the difference in mean crop yield between two types of fertilizer (Type A - Type B). The point estimate is 15 kg, the t* critical value is 2.02, and the standard error is 6 kg. What is the resulting confidence interval?

A) (9, 21) kg

B) (2.88, 27.12) kg

C) (13, 17) kg

D) (3, 27) kg

Correct Answer: B

The interval is calculated as (point estimate) ± t* * SE. This is 15 ± (2.02 * 6) = 15 ± 12.12. The interval is (15 - 12.12, 15 + 12.12), which is (2.88, 27.12).

To satisfy the 'approximately normal' condition for the sampling distribution of the difference of two means, which of the following is generally sufficient?

A) The sum of the two sample sizes is greater than 30.

B) The two sample standard deviations are approximately equal.

C) Both sample sizes are large (e.g., ≥ 30) or both original populations are stated to be normally distributed.

D) The data for both samples are collected using the same method.

Correct Answer: C

The condition that the sampling distribution is approximately normal is checked by ensuring that either the Central Limit Theorem applies (large sample sizes) or the parent populations are normal. This is a standard application of the condition mentioned in the provided content.

A 99% confidence interval for the difference in mean commute times for two cities (City A - City B) is calculated to be (2.1 minutes, 8.5 minutes). Which of the following is a correct interpretation of this interval?

A) There is a 99% probability that the true difference in mean commute times is between 2.1 and 8.5 minutes.

B) We are 99% confident that the sample mean difference is between 2.1 and 8.5 minutes.

C) We are 99% confident that the interval from 2.1 to 8.5 minutes captures the true difference in the population mean commute times.

D) 99% of all sample differences in mean commute times will fall between 2.1 and 8.5 minutes.

Correct Answer: C

The correct interpretation of a confidence interval is a statement of confidence about the interval capturing the true population parameter. The interval is for the difference of two population means, not the sample means.

Based on the 99% confidence interval of (2.1 minutes, 8.5 minutes) for the difference in mean commute times (City A - City B), what conclusion can be drawn?

A) There is no statistically significant difference between the mean commute times of the two cities.

B) We have convincing evidence that the mean commute time in City B is longer than in City A.

C) It is plausible that the mean commute times in the two cities are the same.

D) We have convincing evidence that the mean commute time in City A is longer than in City B.

Correct Answer: D

Since the entire interval contains only positive values, the value of 0 (which would indicate no difference) is not a plausible value for the true difference. This provides convincing evidence that the difference (μA - μB) is positive, meaning the mean commute time in City A is longer.

Holding all other factors constant, how would increasing the confidence level from 90% to 99% affect the margin of error for the difference of two means?

A) The margin of error would decrease.

B) The margin of error would stay the same.

C) The margin of error would increase.

D) The effect cannot be determined without knowing the sample sizes.

Correct Answer: C

A higher confidence level requires a larger t* critical value to create a wider interval that is more likely to capture the true parameter. Since the margin of error is t* times the standard error, a larger t* results in a larger margin of error.

To estimate the difference in mean salaries for male and female professors at a university, a researcher collects salary data from two independent random samples. What is the primary parameter of interest?

A) p1 - p2

B) x1-bar - x2-bar

C) μ1 - μ2

D) s1 - s2

Correct Answer: C

The goal of a confidence interval for the difference of two means is to estimate the difference between the two population means, which are denoted by μ1 and μ2. The value x1-bar - x2-bar is the point estimate used to estimate this parameter.

A confidence interval for the difference of two means is calculated to be (-10.4, 3.2). What does the presence of 0 in this interval imply?

A) It is certain that there is no difference between the two population means.

B) The point estimate for the difference must have been exactly 0.

C) There is a calculation error, as the interval cannot contain both negative and positive values.

D) Zero is a plausible value for the true difference, so there is no convincing evidence of a difference between the two population means.

Correct Answer: D

If the interval of plausible values for the difference (μ1 - μ2) contains 0, it means that a difference of 0 is a possible outcome. Therefore, we cannot conclude that there is a statistically significant difference between the two population means.