AP Statistics Practice Quiz: Constructing a Confidence Interval for a Population Mean

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

According to the properties of t-distributions, why does the use of the sample standard deviation (s) instead of the population standard deviation (sigma) result in a distribution with more area in the tails compared to a normal distribution?

A)Because using 's' introduces additional variability, which is accounted for by the wider tails of the t-distribution.B)Because the sample mean (x-bar) is always a biased estimator of the population mean (mu).C)Because t-distributions are only used for samples that are not randomly selected.D)Because the t-distribution has fewer degrees of freedom than a normal distribution.

All Questions (16)

A) Because using 's' introduces additional variability, which is accounted for by the wider tails of the t-distribution.

B) Because the sample mean (x-bar) is always a biased estimator of the population mean (mu).

C) Because t-distributions are only used for samples that are not randomly selected.

D) Because the t-distribution has fewer degrees of freedom than a normal distribution.

Correct Answer: A

Content point 6 states that when s is used instead of sigma, the resulting t-distribution has more area in the tails. This is because 's' is a statistic that varies from sample to sample, introducing more uncertainty than the fixed parameter sigma, and the t-distribution accounts for this extra variability.

What is the primary effect on the shape of a t-distribution as its degrees of freedom increase?

A) The distribution becomes more skewed to the right.

B) The area in the tails increases, making the distribution wider.

C) The distribution's shape approaches that of a standard normal distribution.

D) The center of the distribution shifts from 0 to 1.

Correct Answer: C

Content point 7 explicitly states that as degrees of freedom increase, the tails of a t-distribution get smaller, and the distribution approaches a normal distribution.

A researcher wants to estimate the mean cholesterol level of a certain population but does not know the population standard deviation. Which statistical procedure is appropriate for constructing a confidence interval in this situation?

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

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

C) A two-sample t-interval for means.

D) A one-sample z-interval for a proportion.

Correct Answer: B

Content point 8 specifies that when the population standard deviation (sigma) is unknown, the appropriate procedure is a one-sample t-interval for a mean.

For a random sample of size n=20 taken from a normally distributed population, which distribution does the statistic (x-bar - mu)/(s/sqrt(n)) follow?

A) A normal distribution with mean 0 and standard deviation 1.

B) A t-distribution with 20 degrees of freedom.

C) A t-distribution with 19 degrees of freedom.

D) A chi-squared distribution with 19 degrees of freedom.

Correct Answer: C

Content point 9 states that this statistic follows a t-distribution with n-1 degrees of freedom. For a sample size of n=20, the degrees of freedom are 20 - 1 = 19.

A study measures the reaction times of 25 participants on a task both before and after drinking coffee. To construct a confidence interval for the mean change in reaction time, how should the data be treated?

A) As two independent samples of 25 reaction times each.

B) As a single sample of the 50 total reaction times.

C) As a single sample of the 25 differences in reaction times.

D) As a proportion of participants whose reaction time improved.

Correct Answer: C

This is a matched pairs design. Content point 10 states that matched pairs data can be analyzed as a single sample of differences.

When constructing a confidence interval for a population mean, what value serves as the point estimate?

A) The sample standard deviation, s.

B) The population standard deviation, sigma.

C) The sample mean, x-bar.

D) The sample size, n.

Correct Answer: C

Content point 15 clearly identifies the sample mean, x-bar, as the point estimate for a population mean.

In the formula for a one-sample t-interval, what does the quantity s/sqrt(n) represent?

A) The margin of error.

B) The standard error of the sample mean.

C) The critical value.

D) The point estimate of the population mean.

Correct Answer: B

Content point 13 defines the standard error for a sample mean as s/sqrt(n).

A random sample of 16 energy bars has a mean of 250 calories and a sample standard deviation of 12 calories. Using a critical value of t* = 2.131, what is the 95% confidence interval for the true mean number of calories?

A) 250 ± 2.131 * (12 / 16)

B) 250 ± 2.131 * (16 / sqrt(12))

C) 12 ± 2.131 * (250 / sqrt(16))

D) 250 ± 2.131 * (12 / sqrt(16))

Correct Answer: D

Using the formula from content point 16, the confidence interval is x-bar ± t*(s/sqrt(n)). Plugging in the values gives 250 ± 2.131 * (12 / sqrt(16)).

Which of the following represents the general formula for the margin of error for a one-sample t-interval?

A) t* * (s / n)

B) x-bar / sqrt(n)

C) t* * (s / sqrt(n))

D) s / sqrt(n)

Correct Answer: C

Content point 14 states that the margin of error for a one-sample t-interval is the critical value t* times the standard error (s/sqrt(n)).

Which of the following are the two primary conditions that must be verified before constructing a confidence interval to estimate a population mean?

A) The population must be normal and the population standard deviation must be known.

B) The sample must be independent (random) and the sampling distribution of the mean must be approximately normal.

C) The sample size must be greater than 10% of the population and the data must be categorical.

D) The number of successes and failures must both be at least 10.

Correct Answer: B

Content point 11 states that to estimate a population mean, one must check for independence and that the sampling distribution is approximately normal. These are the core conditions for inference for a mean.

The general formula for a confidence interval for a population mean is given by x-bar ± t*(s/sqrt(n)). What does the 'x-bar' component represent?

A) The margin of error

B) The standard error

C) The critical value

D) The point estimate

Correct Answer: D

Content point 15 identifies the sample mean, x-bar, as the point estimate for the population mean. The confidence interval is built around this point estimate.

A 90% confidence interval for the mean score on a standardized test is calculated to be (75.2, 81.8). What is the margin of error for this interval?

A) 6.6

B) 78.5

C) 3.3

D) Cannot be determined from the information given.

Correct Answer: C

The confidence interval is centered on the point estimate (x-bar), and its total width is twice the margin of error. The width is 81.8 - 75.2 = 6.6. The margin of error is half the width, so 6.6 / 2 = 3.3. This is derived from the structure of the interval in content point 16: point estimate ± margin of error.

A researcher is conducting a matched pairs experiment to test a new fertilizer. The growth of 30 plants is measured before and after the application. When verifying conditions for a t-interval, what specific data should be checked for approximate normality?

A) The sample of 30 'before' measurements.

B) The sample of 30 'after' measurements.

C) Both the 'before' and 'after' samples independently.

D) The sample of 30 differences between 'after' and 'before' measurements.

Correct Answer: D

Content points 3 and 10 together imply that for matched pairs, the conditions for a one-sample t-interval are applied to the single sample of differences. Therefore, the distribution of the sample differences must be checked for approximate normality.

The critical value t* is a necessary component for calculating the margin of error. How is the value of t* typically determined?

A) By calculating it directly from the sample mean and sample standard deviation.

B) By using a table or technology, based on the confidence level and degrees of freedom.

C) It is always a fixed value, such as 1.96, regardless of sample size.

D) By dividing the sample mean by the sample size.

Correct Answer: B

Content point 12 states that the critical value t* can be found using a table or technology. This process requires knowing the desired confidence level and the degrees of freedom (which is based on sample size).

Which of the following is the correct structure for a one-sample t-interval for a population mean?

A) s ± t*(x-bar/sqrt(n))

B) x-bar ± t*(s/sqrt(n))

C) t* ± x-bar*(s/sqrt(n))

D) n ± t*(s/sqrt(x-bar))

Correct Answer: B

Content point 16 provides the explicit formula for the confidence interval for a population mean as x-bar ± t*(s/sqrt(n)).

Which of the following scenarios would require the use of a one-sample t-interval procedure for a population mean, including the possibility of it being for matched pairs?

A) Estimating the proportion of voters who favor a certain candidate.

B) Comparing the mean test scores of two independent groups of students who used different study methods.

C) Estimating the mean improvement in fuel efficiency for a fleet of cars after a new type of tire is installed, where efficiency is measured on each car before and after the change.

D) Estimating the mean salary of CEOs when the population standard deviation of all CEO salaries is known.

Correct Answer: C

Content point 2 discusses identifying an appropriate confidence interval procedure for a population mean, including for matched pairs. Option C is a classic matched pairs scenario (before/after on the same cars), which is analyzed using a one-sample t-interval on the differences. The other options describe procedures for proportions, two-sample means, or a z-interval (since sigma is known).