Write the unpooled standard error for x_1 - x_2. [1 point]

College Board AP 2022 (style)-style practice: Section II (free response). Independent random samples of two fertilizers give plant heights: fertilizer A, n_1 = 30, x_1 = 42 cm, s_1 = 6; fertilizer B, n_2 = 35, x_2 = 38 cm, s_2 = 7. (a) Check the conditions for a two-sample t-interval for _1 - _2. (b) Construct a 95\% confidence interval (use t^* = 2.00). (c) Justify in context whether there is convincing evidence the mean heights differ.

A 4-point two-sample mean interval. (a) (1 point) Random and independent: two independent random samples. Normal/large: n_1 = 30 and n_2 = 35, both 30, so the CLT applies. 10\%: each sample plausibly under 10\% of its population. (b) (2 points) Difference = 42 - 38 = 4. SE = 6^230 + 7^235 = sqrt(1.2 + 1.4) = sqrt(2.6) ≈ 1.612. Interval = 4 2.00(1.612) = 4 3.225 = (0.78,\ 7.22) cm. (c) (1 point) The interval (0.78, 7.22) excludes 0, so 0 is not a plausible difference; there is convincing evidence the mean heights differ (fertilizer A higher). Markers reward conditions for both samples, the unpooled standard error, the t-interval, and the zero-check conclusion.

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How do you construct a confidence interval for the difference between two population means?

Topic 7.6 Confidence Intervals for the Difference of Two Means: check the conditions and construct a two-sample t-interval for the difference between two population means, including the paired case, using the unpooled standard error.

A focused answer to AP Statistics Topic 7.6, on building a two-sample t-interval for the difference of two population means and distinguishing it from a paired (one-sample) interval, with a full worked interval.

Generated by Claude Opus 4.811 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
The two-sample interval and its standard error
Conditions, doubled
Paired versus two-sample: the key design distinction
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What this topic is asking

The College Board (Topic 7.6) wants you to construct a confidence interval for the difference of two means $\mu_1 - \mu_2$ : check the conditions for both samples, use the unpooled two-sample standard error, and recognize when data are paired (analyzed as a one-sample interval on the differences).

The two-sample interval and its standard error

The point estimate is the observed difference $\bar{x}_1 - \bar{x}_2$ . The standard error follows the Topic 5.8 rule, add the two sample-mean variances $s_1^2/n_1$ and $s_2^2/n_2$ , then square-root, because independent variances add. The degrees of freedom for two-sample $t$ are awkward (the Welch formula), so technology computes them; for hand work, a conservative choice is the smaller of $n_1 - 1$ and $n_2 - 1$ . AP grading accepts either the calculator $df$ or the conservative $df$ .

Conditions, doubled

Everything in the one-sample shape check is applied to each group, plus a between-sample independence requirement. With small samples you must justify normality from a graph for both groups.

Paired versus two-sample: the key design distinction

The single biggest decision in this topic is paired or independent. If each value in one group is naturally matched to a value in the other, two measurements on the same subject (before/after), or matched pairs (twins, left/right), the data are paired. Then you compute one difference per pair and run a one-sample t-interval on those differences: $\bar{x}_d \pm t^{*}\dfrac{s_d}{\sqrt{n}}$ with $df = n - 1$ , where $n$ is the number of pairs. Treating paired data as two independent samples is a serious error: it ignores the pairing that the design used to control variability, and uses the wrong standard error. If the two groups are separate, unrelated sets of units, the design is independent and the two-sample interval applies. Reading the design before choosing the procedure is itself an examined skill.

Two-sample t-interval for a difference in means

Independent random samples compare wait times (minutes) at two clinics. Clinic 1: $n_1 = 40$ , $\bar{x}_1 = 24$ , $s_1 = 8$ . Clinic 2: $n_2 = 50$ , $\bar{x}_2 = 20$ , $s_2 = 6$ . Construct and interpret a $95\%$ confidence interval for $\mu_1 - \mu_2$ (use $t^{*} = 2.01$ ).

step 1 Conditions

Random and independent: two independent random samples. Normal/large: $n_1 = 40$ and $n_2 = 50$ , both $\ge 30$ , so the CLT applies. $10\%$ : each sample plausibly under $10\%$ of its clinic's patients. Conditions met.

step 2 Point estimate and unpooled standard error

$\bar{x}_1 - \bar{x}_2 = 24 - 20 = 4$ minutes.

SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{64}{40} + \frac{36}{50}} = \sqrt{1.6 + 0.72} = \sqrt{2.32} \approx 1.523.

step 3 Margin of error and interval

Margin of error $= t^{*} \cdot SE = 2.01 \times 1.523 \approx 3.061$ .

4 \pm 3.061 = (0.94,\ 7.06) \text{ minutes}.

step 4 Interpret in context

We are $95\%$ confident the difference in mean wait times (clinic 1 minus clinic 2) is between $0.94$ and $7.06$ minutes. Because the interval excludes $0$ , there is convincing evidence the mean wait times differ, with clinic 1 longer.

Try this

Q1. Write the unpooled standard error for $\bar{x}_1 - \bar{x}_2$ . [1 point]

Cue. $SE = \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$ (add the two sample-mean variances, then root).

Q2. A study records each athlete's time on two track surfaces. Two-sample or paired? [1 point]

Cue. Paired (same athlete on both surfaces); analyze the differences with a one-sample t-interval.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2018 (style)1 marksSection I (multiple choice). A study measures each subject before and after a treatment and analyzes the differences. The appropriate procedure is (A) a two-sample t-interval (B) a one-sample t-interval on the differences (paired) (C) a two-proportion z-interval (D) a chi-square test

Show worked answer →

The correct answer is (B).

Before/after measurements on the same subjects are paired data. The correct procedure analyzes the single list of differences with a one-sample t-interval (a paired t-interval), not a two-sample procedure.

(A) is for independent samples, which these are not. (C) is for proportions. (D) is for categorical counts.

AP 2022 (style)4 marksSection II (free response). Independent random samples of two fertilizers give plant heights: fertilizer A,

n_1 = 30

\bar{x}_1 = 42

cm,

s_1 = 6

; fertilizer B,

n_2 = 35

\bar{x}_2 = 38

cm,

s_2 = 7

. (a) Check the conditions for a two-sample t-interval for

\mu_1 - \mu_2

. (b) Construct a

95\%

confidence interval (use

t^{*} = 2.00

). (c) Justify in context whether there is convincing evidence the mean heights differ.

Show worked answer →

A 4-point two-sample mean interval.

(a) (1 point) Random and independent: two independent random samples. Normal/large: $n_1 = 30$ and $n_2 = 35$ , both $\ge 30$ , so the CLT applies. $10\%$ : each sample plausibly under $10\%$ of its population.
(b) (2 points) Difference $= 42 - 38 = 4$ . $SE = \sqrt{\dfrac{6^2}{30} + \dfrac{7^2}{35}} = \sqrt{1.2 + 1.4} = \sqrt{2.6} \approx 1.612$ . Interval $= 4 \pm 2.00(1.612) = 4 \pm 3.225 = (0.78,\ 7.22)$ cm.
(c) (1 point) The interval $(0.78, 7.22)$ excludes $0$ , so $0$ is not a plausible difference; there is convincing evidence the mean heights differ (fertilizer A higher).

Markers reward conditions for both samples, the unpooled standard error, the t-interval, and the zero-check conclusion.

Related dot points

Sources & how we know this

AP Statistics Course and Exam Description — College Board (2020)