A 95\% interval for _1 - _2 is (-7.0, -2.0). What does it say? [1 point]

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How do you use a confidence interval for a difference of two means to justify a claim?

Topic 7.7 Justifying a Claim About the Difference of Two Means Based on a Confidence Interval: use a two-sample (or paired) mean interval to judge whether the means differ and to assess claims about the size and direction of the difference.

A focused answer to AP Statistics Topic 7.7, on using a two-sample or paired mean confidence interval to judge whether two means differ and to assess claims about the size and direction of the difference, with worked justifications.

Generated by Claude Opus 4.89 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 zero-check for two means
Judging size-of-difference claims
Confidence level, sample size, and decisiveness
Try this

What this topic is asking

The College Board (Topic 7.7) wants you to use a two-sample (or paired) mean interval to justify a claim: judge whether the means differ (does the interval contain $0$ ?), and evaluate claims about the size and direction of the difference, accounting for confidence level and sample size.

The zero-check for two means

This is the same plausible-value reasoning as for a single mean, with the relevant null value being $0$ (equal means). The sign of an interval that excludes $0$ gives the direction of the difference, so always state the subtraction order ("group $1$ minus group $2$ "). For paired data, the interval is for the mean difference $\mu_d$ , and the zero-check asks whether $\mu_d = 0$ (no average change) is plausible.

Judging size-of-difference claims

A magnitude claim, "the new method lowers the mean by at least $3$ ," must be judged against the entire interval, not the point estimate. If part of the interval falls short of the claimed size (here, includes values greater than $-3$ ), then a smaller effect is still plausible and the claim is not established, even if the observed difference meets it. Only if the whole interval satisfies the claim is it supported. As always, the interval, the set of plausible differences, carries the justification, not the single observed difference. Distinguishing "is there a difference?" from "is the difference at least this big?" is a recurring free-response discriminator.

Confidence level, sample size, and decisiveness

A higher confidence level widens the interval, which can pull a once-decisive interval back across $0$ and weaken a difference conclusion; a lower level narrows it (more decisive, correct less often). A larger sample in either group narrows the interval by shrinking the standard error, sharpening conclusions about both the existence and size of a difference. These trade-offs mirror the one-sample case; exam questions ask you to predict the direction of each change and explain the mechanism.

Justifying a two-mean difference claim

A $95\%$ confidence interval for $\mu_A - \mu_B$ , the difference in mean test scores (method A minus method B), is $(1.2,\ 6.8)$ . (a) Is there convincing evidence the methods differ? (b) A teacher claims method A raises the mean by at least $5$ points; assess this. (c) Predict the effect of a larger sample.

step 1 Zero-check (part a)

The interval $(1.2, 6.8)$ lies entirely above $0$ , so $0$ is not a plausible difference. There is convincing evidence the methods differ, with method A producing a higher mean score (the interval is positive and the subtraction is A minus B).

step 2 Assess the magnitude claim (part b)

"At least $5$ points" means $\mu_A - \mu_B \ge 5$ . The interval includes plausible differences below $5$ (for example $2$ ), so a smaller gain is still plausible. The interval does not establish that method A raises the mean by at least $5$ points.

step 3 Effect of a larger sample (part c)

A larger sample shrinks the standard error, narrowing the interval around the same center of about $4$ . The narrower interval can exclude more values, making the conclusions about both the existence and the size of the difference more decisive.

step 4 Summary

There is evidence of a difference (A higher) at $95\%$ , but not enough to confirm a gain of at least $5$ points; a larger sample would sharpen the size conclusion.

Try this

Q1. A $95\%$ interval for $\mu_1 - \mu_2$ is $(-7.0, -2.0)$ . What does it say? [1 point]

Cue. Entirely below $0$ : convincing evidence $\mu_1 < \mu_2$ (group $1$ mean is smaller).

Q2. For paired data, what does the zero-check ask? [1 point]

Cue. Whether $\mu_d = 0$ (no average difference) is plausible; if $0$ is outside the interval, there is evidence of a real mean change.

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 2017 (style)1 marksSection I (multiple choice). A

95\%

confidence interval for

\mu_1 - \mu_2

(-1.5,\ 4.0)

. Based on this interval, there is (A) convincing evidence

\mu_1 > \mu_2

(B) convincing evidence

\mu_1 < \mu_2

Show worked answer →

The correct answer is (C).

The interval $(-1.5, 4.0)$ contains $0$ , so $0$ (no difference) is a plausible value of $\mu_1 - \mu_2$ ; there is no convincing evidence the two means differ.

(A) and (B) require the whole interval above or below $0$ . (D) overstates: an interval never proves equality.

AP 2021 (style)4 marksSection II (free response). A

90\%

confidence interval for

\mu_{\text{new}} - \mu_{\text{old}}

, the difference in mean processing time (seconds) for a new versus old system, is

(-4.5,\ -0.5)

. (a) Justify in context whether there is convincing evidence the new system is faster (smaller mean time). (b) An engineer claims the new system cuts mean time by at least

3

seconds. Justify whether the interval supports this. (c) Explain how a larger sample would change the interval and the justifications.

Show worked answer →

A 4-point justification question on a difference.

(a) (2 points) The entire interval $(-4.5, -0.5)$ lies below $0$ , so $0$ is not a plausible difference; there is convincing evidence that $\mu_{\text{new}} < \mu_{\text{old}}$ , i.e. the new system has a smaller mean processing time (is faster).
(b) (1 point) "Cuts by at least $3$ seconds" means $\mu_{\text{new}} - \mu_{\text{old}} \le -3$ . The interval includes values above $-3$ (such as $-1$ ), so a smaller reduction is still plausible; the interval does not establish a cut of at least $3$ seconds.
(c) (1 point) A larger sample shrinks the standard error, narrowing the interval; this could exclude more values, making any conclusion about the size of the difference more decisive (without changing the center on average).

Markers reward the zero-check for part (a), reading the whole interval against $-3$ for part (b), and linking larger samples to a narrower interval.

Related dot points

Sources & how we know this

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