AP Statistics t Intervals and t Tests for a Mean Worksheet (PDF)

AP Statistics math worksheet: t Intervals and t Tests for a Mean – Maths PDF

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This inference one mean worksheet for AP Statistics covers t intervals and t tests for a mean (Common Core AP STAT 7.1, AP STAT 7.2, AP STAT 7.3, AP STAT 7.4, AP STAT 7.5, HSS-IC.B.4), with a free PDF to print.

These 20 problems practice AP Statistics topics 7.1 to 7.5. You will find t critical values and standard errors, build and interpret one-sample t intervals, run one-sample t tests and matched-pairs tests, and check conditions with graphs.

AP Statistics · Common Core State Standards: AP STAT 7.1, AP STAT 7.2, AP STAT 7.3, AP STAT 7.4, AP STAT 7.5, HSS-IC.B.4

Inference one mean worksheet: exercises 1 to 10

Exercise 1 – t versus normal

The t distribution has heavier tails than the normal; it approaches the normal as degrees of freedom grow.

Describe two differences between the t distribution with 4 degrees of freedom and the standard normal distribution.

Exercise 2 – Degrees of freedom and t*

Critical values t* decrease toward z* as df grows.

Find df and the critical value t* for a 95% interval from a sample of size 21, and for a 99% interval from a sample of size 31.

Exercise 3 – Standard error

A random sample of 25 sleep times has mean 7.2 hours and standard deviation 1.4 hours. Find the standard error of the mean.

Exercise 4 – Compute an interval

Using the sleep data (n = 25, mean 7.2, s = 1.4), find a 95% confidence interval for the mean sleep time of all students at the school.

Exercise 5 – Interpret the interval

95% t interval: (38.96, 41.75) hours.

Interpret the battery interval (38.96, 41.75) in context.

Exercise 6 – Check the Normal condition

Ten wait times in minutes.

A random sample of 10 wait times (minutes) is graphed. Is a t interval appropriate?

Exercise 7 – Write hypotheses

A pizza shop claims its mean delivery time is 30 minutes. A customer group thinks it is longer. Write H0 and Ha and define the parameter.

Exercise 8 – Compute t

A random sample of 40 deliveries has mean 33.2 minutes and standard deviation 6.5 minutes. Compute the t statistic for H0: mu = 30.

Exercise 9 – p-value

Right-tail p-value.

For the delivery data, df = 39 and Ha: mu > 30. Find the p-value.

Exercise 10 – Conclusion

State a conclusion for the delivery test at alpha = 0.05 (p-value 0.0017).

Exercises 11 and up

Exercise 11 – Two-sided t test

Body temperatures: a random sample of 36 adults has mean 98.2 degrees F and standard deviation 1.1 degrees F. Test whether the true mean differs from 98.6 degrees F at alpha = 0.05.

Exercise 12 – Interval and test

A 95% t interval for mu is (41.2, 46.8). A two-sided test of H0: mu = 40 at alpha = 0.05 would lead to what conclusion?

Exercise 13 – Matched pairs differences

Eight patients’ cholesterol (mg/dL) before and after a diet: before 182, 195, 170, 188, 201, 176, 190, 185; after 178, 190, 168, 180, 199, 172, 185, 183. Find the differences (before minus after), their mean, and their standard deviation.

Exercise 14 – Matched pairs interval

Differences (before minus after) in mg/dL.

Construct a 95% interval for the true mean reduction (before minus after).

Exercise 15 – Matched pairs test

Test whether the diet lowers cholesterol on average (H0: mu_d = 0, Ha: mu_d > 0) using the differences above.

Exercise 16 – Effect of n

Two samples have the same mean 50 and s = 10, one with n = 16 and one with n = 64. Compare the 95% margins of error.

Exercise 17 – Read output

Output for a one-sample t procedure.

The output shows summary statistics for a random sample of completion times (minutes). Verify the standard error and find a 90% interval for the mean.

Exercise 18 – Outlier warning

A random sample of 12 salaries (in thousands) includes one value of 400 while the others are between 40 and 70. Explain why a t interval for the mean is questionable.

Exercise 19 – Left-tailed test

A company claims its batteries last at least 25 hours. A random sample of 50 batteries has mean 24.1 hours and s = 3.9 hours. Test at alpha = 0.05 whether the mean is less than 25.

Exercise 20 – Multi-step

A random sample of 30 apartments has a mean rent of $1,350 and s = $180. (a) Find a 99% confidence interval for the mean rent. (b) Is $1,300 plausible?

Key facts behind this inference one mean worksheet

Key facts

t distribution: bell-shaped, centered at 0, heavier tails; df = n – 1.
Interval: x-bar +/- t* s/sqrt(n); test: t = (x-bar – mu0)/(s/sqrt(n)).
Conditions: random, 10% condition, and Normal/Large Sample (population normal, n >= 30, or no strong skew or outliers).
As df grows, t* approaches z*.
Matched pairs: take differences and do a one-sample t procedure.

How to use this inference one mean 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 inference one mean worksheet as a PDF with the button below and keep it with your notes.

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