AP Statistics · Exam guide
AP Statistics Inference for Means Explained: The One-Sample t Test and Checking Conditions
Inference for means looks like inference for proportions, with two differences: you use a t distribution, and the condition about normality takes real thought. Graders check both. The data below are made up for practice.
A made-up claim
A company claims its batteries last 10 hours on average. A consumer group tests a random sample of 25 batteries and finds a mean life of 9.6 hours with a standard deviation of 1.0 hour. The sample data are roughly symmetric with no outliers. Is there convincing evidence that the true mean is less than 10 hours?
| Step | Work |
|---|---|
| State | H0: mu = 10. Ha: mu < 10, where mu is the true mean battery life, in hours, of all batteries of this type. |
| Plan | One-sample t test for a mean. Random sample. The sample is less than 10 percent of all batteries. The sample size is 25, which is less than 30, so check normality: the sample data are roughly symmetric with no outliers, so the population is reasonably normal. |
| Do | t = (9.6 - 10) / (1.0 / the square root of 25) = -0.4 / 0.2 = -2.0, with 24 degrees of freedom. The p-value is about 0.028. |
| Conclude | The p-value of 0.028 is less than 0.05, so we reject H0. There is convincing evidence that the true mean battery life is less than 10 hours. |
Why the normal condition matters
With a sample of 30 or more, the sampling distribution of the mean is close to normal even if the population is skewed. With a smaller sample, you must show that the data do not suggest a skewed or outlier-heavy population. Look at a graph of the data, such as a dotplot or boxplot, and describe what you see.
With n = 25, that is not the rule. It also gives no look at the data.
It uses the sample size and describes the data.
Choosing the procedure
| Situation | Procedure |
|---|---|
| One sample, one quantitative variable | One-sample t test or interval |
| Two independent groups, one quantitative variable | Two-sample t test or interval |
| Two measurements on the same individuals, or matched pairs | Paired t: take differences and do a one-sample t on them |
Common mistakes
- Using z instead of t when the population standard deviation is unknown.
- Using the rule of n of at least 30 when it is not true, or never checking the data.
- Treating paired data as two independent samples.
- Forgetting the degrees of freedom, which is n minus 1 for one sample.
- Concluding without saying what mu means.
Frequently asked questions
When do I use t instead of z?
When you estimate the population standard deviation from the sample, which is almost always the case with means.
What are the degrees of freedom for a one-sample t?
n minus 1.
How do I show the normal condition?
Say the sample size is at least 30, or describe a graph of the data that shows no strong skew and no outliers.
How do I recognize paired data?
Each individual or matched pair gives two measurements, and the question is about the difference between them.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every unit in the AP Statistics course ends with a ten-problem practice set, three multiple choice and seven free response, with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Statistics course Free lessons and lesson list
Other free-response guides:
- AP Statistics FRQ Guide: How Free-Response Answers Are Scored
- AP Statistics Inference for Proportions Explained: Intervals and Tests with a Worked Example
- AP Statistics Chi-Square Test Explained: Homogeneity and Independence with a Worked Example
- AP Statistics Regression Questions Explained: Interpreting Slope, r-Squared and Residuals
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