Topic 7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval: use a one-sample mean interval to judge whether a claimed mean is plausible, and explain how confidence level and sample size affect the interval.

What this topic is asking

The College Board (Topic 7.3) wants you to use a one-sample mean interval to justify a claim about $\mu$: decide whether a claimed value is plausible (inside the interval?), assess directional claims against the whole interval, and explain how confidence level and sample size change the interval and the conclusion.

The plausible-values logic for means

This is the same reasoning as for proportions, now applied to a mean. You do not need a separate significance test to assess one claimed value: check whether it lies inside the interval. A two-sided t-test at $\alpha = 1 - C$ reaches the same verdict, the duality that links Topic 7.3 to the testing topics that follow.

Directional claims need the whole interval

A claim like "$\mu$ is less than $505$" must be checked against the entire interval, not the point estimate $\bar{x}$. If any part of the interval reaches or exceeds $505$, then a mean of $505$ or more is still plausible, so the claim is not established, even when $\bar{x} < 505$. Only if the whole interval lies below $505$ is "$\mu < 505$" supported. The same applies to "at least," "more than," and "between" claims: the interval, the set of plausible means, carries the justification, never the single observed mean. This is one of the most reliable free-response discriminators in the unit.

How confidence level and sample size move the interval

Raising confidence widens the interval (more often correct, less precise), so more claimed values become plausible and fewer can be ruled out, a less decisive justification. A larger sample shrinks the standard error $s/\sqrt{n}$, narrowing the interval (precision improving with $\sqrt{n}$), so more values fall outside and can be excluded, a more decisive justification. A larger sample changes precision, not the center on average, so it does not bias the estimate toward any claim. Questions routinely ask you to predict and explain the direction of these effects.

Try this

Q1. A $90\%$ interval for $\mu$ is $(12.1, 13.9)$. Is a claimed mean of $13$ plausible? [1 point]

• Cue. Yes; $13$ is inside the interval, so it is a plausible value of $\mu$.

Q2. Holding confidence fixed, how does a larger sample affect the ability to rule out a claimed mean? [1 point]

• Cue. It narrows the interval, so more values fall outside and can be ruled out; the justification becomes more decisive.