AP Stats 4.8: Justifying a Claim, Two-Mean CI

What topic 4.8 covers

Topic 4.8 is the interpretation half of the two-sample interval from topic 4.7. You state what the interval means for the difference between two population means, $\mu_1 - \mu_2$, and use it to justify a claim about whether the two means differ. No new arithmetic is introduced.

As with a single mean, a correct interpretation names the difference in means, the response variable, and both populations in context.

Interpreting the interval and the confidence level

Keep the two statements distinct.

Interpret one interval: you are C% confident that the interval from $a$ to $b$ contains the true difference between the two population means, in context and with the order of subtraction stated.

Interpret the confidence level: in repeated random sampling with the same sample sizes from the same two populations, about C% of the intervals built this way would capture the true difference in means. The level describes the method across many samples, not the single interval in front of you. The exact phrasing is the same idea as what 95 percent confidence means.

Both statements should name the response variable and the two populations, so a reader knows exactly which difference the interval estimates and in which direction it was measured.

Using zero to judge a difference

The key move in topic 4.8 is checking whether 0 lies in the interval, because 0 represents no difference between the means.

• If the interval contains 0, then 0 is a plausible value for $\mu_1 - \mu_2$, so there is not convincing evidence that the two population means differ.
• If the interval does not contain 0, then every plausible value of the difference is positive or every value is negative, so there is convincing evidence of a difference, and the sign tells you which mean is larger.

This mirrors a two-sided two-sample t-test at the matching significance level. Confirm intervals with the two-sample t-test calculator.

The width of the interval also carries information. A narrow interval far from 0 points to a difference that is both real and precisely estimated, while a wide interval that straddles 0 leaves the direction of any difference genuinely uncertain. Reporting the interval, not just whether it contains 0, gives the fuller picture graders look for.