AP Statistics · Topic 4.4 · Unit 4

AP Stats 4.4: Setting Up a t-Test for a Mean

By Jude Wallis · Published July 28, 2026

Topic 4.4 sets up a one-sample t-test for a mean when sigma is unknown. The null hypothesis is mu equals mu-naught; the alternative is one-sided or two-sided. For matched pairs the parameter is the mean difference mu-d with null value 0. Then verify the random, 10 percent, and normality conditions.

AP Statistics: Unit 4 (topics 4.4). Topic 4.4 (Setting Up a Test for a Population Mean or Population Mean Difference) in the Fall 2026 AP Statistics course.

What topic 4.4 covers

Topic 4.4 is the setup stage of a significance test about a mean: choose the method, name the parameter, write the hypotheses, and check conditions. You do not compute a test statistic here; that comes in topic 4.5.

Because the population standard deviation $\sigma$ (sigma) is unknown for a quantitative variable, the correct method is a one-sample t-test for a population mean. For a matched pairs design with two dependent samples, you analyze the differences and run a one-sample t-test for the population mean difference.

Writing the hypotheses

State hypotheses in terms of a population parameter, never a sample statistic, and name the response variable and population in context.

For a single mean, the null hypothesis is

H_0: \mu = \mu_0,

where $\mu_0$ is the hypothesized value. The alternative is one of $H_a: \mu < \mu_0$ , $H_a: \mu > \mu_0$ , or $H_a: \mu \ne \mu_0$ . Pick the direction from the question before seeing the data.

For a matched pairs design the parameter is the population mean difference $\mu_d$ , the null hypothesis is $H_0: \mu_d = 0$ , and the alternative is $H_a: \mu_d < 0$ , $H_a: \mu_d > 0$ , or $H_a: \mu_d \ne 0$ . Always state the order of subtraction. For the choice between one-sided and two-sided wording, see null vs alternative hypothesis.

Whichever form you use, define the parameter fully before writing symbols: name the population parameter, the response variable, and the population, all in context. On the free-response section, hypotheses stated only with bare symbols, or stated about a sample statistic such as $\bar{x}$ , lose credit because they describe the sample rather than the population you want to infer about.

Verifying the conditions

A one-sample t-test for a mean or mean difference requires the same three conditions as the matching interval, each checked in context:

Randomization: the data come from a random sample or a randomized experiment.
10% condition: when sampling without replacement, $n \le 0.10N$ , where $N$ is the population size.
Sample data (normality): the population is stated to be approximately normal, or $n \ge 30$ ; if $n < 30$ , the sample should be free from strong skewness and outliers. For matched pairs, apply this to the differences.

If you are unsure whether means, proportions, or a chi-square procedure fits a problem, work through which statistical test to use.

Setting up a one-sample t-test

A bus company advertises a mean commute time of 30 minutes on a route. A rider group suspects the true mean is longer and records a random sample of 16 trips, which show no strong skew or outliers. Identify the procedure, state the hypotheses, and verify the conditions.

The variable, commute time, is quantitative and the population standard deviation is unknown, so use a one-sample t-test for a population mean.
Define the parameter: let $\mu$ be the true mean commute time, in minutes, for all trips on this route.
State the hypotheses. The claim to test is that the mean is longer than advertised, so $H_0: \mu = 30$ and $H_a: \mu > 30$ .
Randomization: the 16 trips are a random sample, so this holds.
10% condition: 16 trips are far fewer than 10% of all trips on the route, so this holds.
Sample data: $n = 16 < 30$ , but the sample shows no strong skew or outliers, so a t procedure is reasonable.

Use a one-sample t-test for the mean commute time with $H_0: \mu = 30$ minutes and $H_a: \mu > 30$ minutes. All three conditions are met, so carrying out the test in topic 4.5 is justified.

Frequently asked questions

Should the alternative hypothesis be one-sided or two-sided?

Decide from the research question before looking at the data. Use a one-sided alternative when the question asks whether the mean is specifically greater than or specifically less than a value. Use a two-sided alternative when the question only asks whether the mean differs from the value. Never pick the direction after seeing which way the sample came out.

How is a matched-pairs test set up differently?

For matched pairs you first compute the difference within each pair to get one sample of differences, then test the mean difference mu-d. The null hypothesis is that mu-d equals 0, meaning no average change, and the alternative reflects the question. Always state the order of subtraction so the sign of the alternative is clear.