AP Statistics · Topic 4.7 · Unit 4

AP Stats 4.7: Confidence Interval for Two Means

By Jude Wallis · Published July 28, 2026

Topic 4.7 builds a two-sample t-interval: x-bar-1 minus x-bar-2, plus or minus t-star times the square root of s-1 squared over n-1 plus s-2 squared over n-2. The parameter is mu-1 minus mu-2, and technology gives the degrees of freedom.

AP Statistics: Unit 4 (topics 4.7). Topic 4.7 (Constructing a Confidence Interval for the Difference Between Two Population Means) in the Fall 2026 AP Statistics course.

What topic 4.7 covers

Topic 4.7 estimates the difference between two population means, $\mu_1 - \mu_2$ , with an interval built from two independent samples. The College Board name is a two-sample t-interval for the difference between population means. You identify the procedure, justify it by conditions, and calculate the interval, standard error, and margin of error.

The point estimate is the observed difference in sample means, $\bar{x}_1 - \bar{x}_2$ . State the parameter with the response variable and both populations in context, and make the order of subtraction clear.

The interval, standard error, and degrees of freedom

The interval is the point estimate plus or minus a margin of error:

(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}.

The standard error of the difference is $SE = \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$ , using the two sample standard deviations, and the margin of error is $t^* \cdot SE$ . Unlike one sample, the degrees of freedom are not a tidy formula: they fall between the smaller of $n_1 - 1$ and $n_2 - 1$ and the value $n_1 + n_2 - 2$ , and on the exam you read the exact df from technology. A safe hand method uses the smaller of $n_1 - 1$ and $n_2 - 1$ , which gives a slightly wider, conservative interval. The two-sample t-test calculator reports both the interval and the technology df.

Conditions to justify the interval

A two-sample t-interval requires three conditions, stated in context:

Randomization: two independent random samples, or a randomized experiment.
10% condition: when sampling without replacement, $n_1 \le 0.10N_1$ and $n_2 \le 0.10N_2$ . This is unnecessary for a randomized experiment.
Sample data (normality): both sample sizes are at least 30, or both populations are stated to be approximately normal. If either sample size is under 30, both sample distributions should be free from strong skewness and outliers.

The interval shares its structure with every other AP interval; see how to calculate a confidence interval and check your work with the confidence interval calculator.

The AP two-sample procedure does not pool the two sample standard deviations, so each group keeps its own $s$ and its own sample size in the standard error. This unpooled interval is what a graphing calculator reports by default, and it is the version expected on the exam, so avoid any formula that combines the two standard deviations into one.

95% two-sample t-interval for a difference of means

Two independent random samples of plants are grown under two fertilizers. Fertilizer 1: $n_1 = 10$ , $\bar{x}_1 = 25$ cm, $s_1 = 4$ cm. Fertilizer 2: $n_2 = 10$ , $\bar{x}_2 = 22$ cm, $s_2 = 3$ cm. Both samples are roughly symmetric with no outliers. Construct a 95% confidence interval for $\mu_1 - \mu_2$ .

Point estimate: $\bar{x}_1 - \bar{x}_2 = 25 - 22 = 3$ cm.
Standard error: $SE = \sqrt{\frac{4^2}{10} + \frac{3^2}{10}} = \sqrt{\frac{16}{10} + \frac{9}{10}} = \sqrt{\frac{25}{10}} = \sqrt{2.5} \approx 1.5811$ cm.
Use the conservative degrees of freedom, the smaller of $10 - 1$ and $10 - 1$ , so $df = 9$ . For 95% confidence, $t^* = 2.262$ .
Margin of error: $t^* \cdot SE = 2.262 \times 1.5811 \approx 3.577$ cm.
Interval: $3 \pm 3.577 = (-0.577,\ 6.577)$ cm.

Using the conservative $df = 9$ , you are 95% confident that the difference in mean plant height (fertilizer 1 minus fertilizer 2) is between about -0.58 and 6.58 cm. Technology, which uses a df between 9 and 18, gives a slightly narrower interval.

Frequently asked questions

What degrees of freedom should I use for a two-sample t-interval?

On the exam, read the degrees of freedom from technology, which uses a formula giving a value between the smaller of the two sample sizes minus one and the sum of the two sample sizes minus two. By hand, using the smaller of the two sample sizes minus one is a valid conservative choice that produces a slightly wider interval.

Is this the same as a matched-pairs interval?

No. A two-sample t-interval compares two independent groups and uses the standard error with s-1 squared over n-1 plus s-2 squared over n-2. A matched-pairs interval comes from dependent, linked measurements, where you first take within-pair differences and then run a one-sample t-interval on those differences.