AP Statistics Inference for the Difference of Two Means Worksheet (PDF)

AP Statistics math worksheet: Inference for the Difference of Two Means – Maths PDF

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This two means worksheet for AP Statistics covers inference for the difference of two means (Common Core AP STAT 7.6, AP STAT 7.7, AP STAT 7.8, AP STAT 7.9, HSS-IC.B.5), with a free PDF to print.

These 20 problems practice AP Statistics topics 7.6 to 7.9. You will compute standard errors for a difference of means, build and interpret two-sample t intervals, run two-sample t tests, and decide between two-sample and paired designs.

AP Statistics · Common Core State Standards: AP STAT 7.6, AP STAT 7.7, AP STAT 7.8, AP STAT 7.9, HSS-IC.B.5

Two means worksheet: exercises 1 to 10

Exercise 1 – Parameters and hypotheses

Scores in two independent classes taught with different methods.

Using the table, define the parameters and write hypotheses to test whether the new method gives a higher mean score.

Exercise 2 – Difference and standard error

For the two classes (A: n = 35, mean 78.4, s = 9.2; B: n = 40, mean 72.1, s = 11.5), find the difference of means and the standard error.

Exercise 3 – Degrees of freedom

For the same data, find the conservative degrees of freedom and compare it with the calculator value 72.5.

Exercise 4 – Confidence interval

95% interval for mu_A – mu_B: (1.53, 11.07); 0 is not included.

Find a 95% confidence interval for mu_A – mu_B.

Exercise 5 – Interpret the interval

Interpret the interval (1.531, 11.069) in context.

Exercise 6 – Check conditions

Boxplots of both classes: no outliers and roughly symmetric.

Check the conditions for a two-sample t procedure, given independent random samples of 35 and 40 students from large schools and the boxplots shown.

Exercise 7 – Compute t

Compute the t statistic for H0: mu_A – mu_B = 0 using the school data.

Exercise 8 – p-value

Two-sided p-value for t = 2.63: 0.0104.

Find the two-sided p-value for t = 2.63 with df = 72.5.

Exercise 9 – Conclusion

State the conclusion at alpha = 0.05 (p-value 0.0104).

Exercise 10 – Experiment with caffeine

Volunteers were randomly assigned to caffeine (n = 12, mean 25.4 correct, s = 4.8) or a placebo (n = 14, mean 22.1, s = 5.2) on a memory test. Test whether caffeine increases the mean number correct at alpha = 0.05.

Exercises 11 and up

Exercise 11 – From raw data

Group A.

Two independent random groups of 8 plants were grown with different fertilizers (growth in cm). Fertilizer 1: 55, 61, 58, 66, 52, 60, 63, 57. Fertilizer 2: 48, 54, 50, 61, 45, 53, 56, 49. Find the means and standard deviations, then a 95% interval for mu1 – mu2.

Exercise 12 – Paired or independent?

Decide whether to use a paired or a two-sample procedure: (a) 30 runners’ times before and after a month of training; (b) mean salaries of 40 randomly selected teachers in two districts; (c) 20 pairs of twins, one twin randomly assigned to each of two diets.

Exercise 13 – Wrong procedure

Choose the procedure by the design.

A student uses a two-sample t procedure for scores of the same 25 students before and after a workshop. Explain the error and the correct procedure.

Exercise 14 – Interval and test

A 95% interval for mu1 – mu2 is (-1.2, 6.8). What does this say about a two-sided test of H0: mu1 = mu2 at alpha = 0.05?

Exercise 15 – Homework hours

Random samples: 45 boys average 2.9 hours of homework per week (s = 1.4) and 50 girls average 2.4 hours (s = 1.1). Test whether the true means differ at alpha = 0.05.

Exercise 16 – Effect of sample size

Two studies find the same difference of means (4 points) with s = 10 in each group. Compare the standard errors when n = 25 per group and n = 100 per group.

Exercise 17 – Outlier problem

Group 1 has mean 50, s = 5, n = 15. Group 2 has mean 56, s = 25, n = 15 because of one value of 150. Explain why a two-sample t interval for the difference may be unreliable.

Exercise 18 – 99 percent interval

Random samples: 60 adults in city A spend a mean of 34.5 minutes (s = 6.0) commuting and 55 adults in city B spend 31.0 minutes (s = 5.5). Find a 99% interval for mu_A – mu_B.

Exercise 19 – Multi-step

A randomized experiment compares two study plans: Plan 1 (n = 30, mean gain 12.5, s = 4.0) and Plan 2 (n = 28, mean gain 10.2, s = 3.5). (a) Test Ha: mu1 > mu2 at alpha = 0.05. (b) Would a 90% interval agree?

Exercise 20 – Describe the difference

The 95% interval for the mean difference in reaction time (men minus women) is (-8, 21) ms. Write a conclusion in context.

Key facts behind this two means worksheet

Key facts

Parameter: mu1 – mu2; statistic: x-bar1 – x-bar2.
SE = sqrt(s1^2/n1 + s2^2/n2); df from the calculator, or min(n1 – 1, n2 – 1) conservatively.
Interval: (x-bar1 – x-bar2) +/- t* SE; test: t = (x-bar1 – x-bar2)/SE with H0: mu1 – mu2 = 0.
Conditions: independent random samples or random assignment; 10%; each group normal or n >= 30.
Same subjects measured twice or matched pairs: use a one-sample t on the differences.

How to use this two means worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the two means worksheet as a PDF with the button below and keep it with your notes.

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