Inference for Means Worksheet with Answers – AP Statistics (Free PDF)

Problem 1 Standard Error of a Mean ★★★

A random sample of 25 fitness-club members has a mean of 148 minutes of weekly cardio with a standard deviation of 30 minutes. Find the standard error of the sample mean.

Problem 2 Finding a Critical Value ★★★

Find the critical value t* for (a) a 95% interval with n = 12, (b) a 99% interval with n = 20, (c) a 90% interval with n = 8.

Problem 3 A One-Sample t Interval ★★★

A random sample of 16 bags of coffee beans has a mean weight of 4.62 pounds with a standard deviation of 0.85 pound. Assuming the conditions are met, find a 95% confidence interval for the mean weight of all bags.

Problem 4 Checking Conditions ★★★

A random sample of 10 customers was timed at the checkout. The dotplot shows the waiting times.

Is it appropriate to use a one-sample t interval for the mean waiting time? Explain.

Problem 5 Interpreting an Interval ★★★

A 95% confidence interval for the mean weekly screen time of students at a school is (18.47, 24.33) hours. Interpret the interval and the confidence level.

Problem 6 Writing Hypotheses for Means ★★★

Write the hypotheses for each claim, defining μ.

1. A bakery says its loaves weigh 450 grams on average; an inspector suspects they weigh less.
2. A school claims students sleep 8 hours on average on school nights; a nurse thinks that is not true.
3. A battery maker says its batteries last 12 hours on average; a competitor suspects they last more.

Problem 7 Test Statistic and p-Value ★★★

A random sample of 22 loaves has a mean weight of 448.2 g with a standard deviation of 4.1 g. Test H₀: μ = 450 against H_a: μ < 450. Find the test statistic and p-value.

Problem 8 Two-Sided t Test ★★★

A school claims students sleep 8 hours on average on school nights. A random sample of 30 students has a mean of 7.4 hours and a standard deviation of 1.3 hours. Test H₀: μ = 8 against H_a: μ ≠ 8 at α = 0.05, assuming the conditions are met.

Problem 9 A Claim from an Interval ★★★

A 95% confidence interval for a mean (random sample of 24 students, x̄ = 6.1, s = 1.5) is computed. Does the interval give convincing evidence against the claim that the mean is 7 hours?

Problem 10 Sample Size and Interval Width ★★★

For s = 12, compare the margins of error of 95% t intervals for n = 9 and n = 36. Which is wider and why?

Problem 11 Paired or Two-Sample ★★★

Decide whether each study calls for a paired procedure or a two-sample procedure.

1. Fifteen runners’ times are recorded with and without a new insole.
2. Heights of 40 randomly chosen women are compared with heights of 35 randomly chosen men.
3. Twenty twin pairs, one twin randomly given tutoring and the other not, are compared on a final test.

Problem 12 Paired t Interval ★★★

Eight randomly chosen students took a quiz before and after a review session. The dotplot shows the changes (after minus before).

Find a 95% confidence interval for the mean change.

Problem 13 Paired t Test ★★★

Ten randomly selected drivers had their braking reaction time (milliseconds) measured while texting and while not texting. The differences (texting minus not texting) are 12, 18, −7, 22, 5, 9, 25, −4, 19, 1. Test whether texting increases the mean reaction time, using α = 0.05.

Problem 14 Two-Sample t Interval ★★★

Independent random samples of mice on two diets gave these weight gains (grams): Diet A, n = 28, mean 36.4, standard deviation 5.2; Diet B, n = 31, mean 32.9, standard deviation 6.1. Assuming the conditions are met, find a 95% confidence interval for μ_A − μ_B.

Problem 15 Two-Sample t Test ★★★

Students were randomly assigned to two teaching methods. The boxplots show the scores; summary statistics are Method 1: n = 30, mean 82.4, s = 9.6; Method 2: n = 28, mean 77.1, s = 11.2.

Do the data provide convincing evidence that the mean scores differ? Use α = 0.05.

Problem 16 Interval and Test Agree ★★★

For the teaching-method data in Problem 15, a 95% confidence interval for μ₁ − μ₂ is calculated. Find it and explain how it agrees with the test result.

Problem 17 Selecting a Procedure ★★★

Name the appropriate inference procedure for each situation.

1. Compare the proportion of voters favoring a bill in two states, using independent random samples.
2. Estimate the mean number of hours worked by all students in a district using one random sample.
3. Test whether a new drug lowers blood pressure using the same 30 patients before and after treatment.
4. Test whether the distribution of favorite pet is the same among three age groups.

Problem 18 A Full Two-Sample Test ★★★

Two independent random samples of students took a timed puzzle test. Scores for Group A (n = 9): 72, 65, 80, 74, 69, 77, 70, 83, 68. Scores for Group B (n = 10): 63, 70, 61, 68, 59, 66, 72, 64, 60, 67. Is there convincing evidence that the mean scores differ? Use α = 0.05 and complete all steps.

Problem 19 Confidence Level and Two-Sided Tests ★★★

A 99% confidence interval for a mean is (41.2, 48.6). (a) What can you conclude about H₀: μ = 50 versus H_a: μ ≠ 50 at α = 0.01? (b) What can you conclude about H₀: μ = 45 at α = 0.01? (c) Would a 90% interval from the same data be wider or narrower?

Problem 20 Why Pairing Matters ★★★

Eight randomly chosen adults had their resting pulse measured before and after a month of walking. Before: 62, 71, 55, 68, 74, 59, 80, 66. After: 58, 65, 54, 60, 66, 56, 71, 61. (a) Analyze the data as paired and find the t statistic and two-sided p-value. (b) Analyze incorrectly as two independent samples. (c) Compare the two analyses.