AP Stats 4.3: Justifying a Claim from a Mean CI

What topic 4.3 covers

Topic 4.3 is the interpretation half of the confidence interval you built in topic 4.2. You do no new arithmetic here. Instead you learn to state what the interval means, use it to judge a claim about the population mean $\mu$ (mu), and describe how sample size, confidence level, margin of error, and width relate.

These are the skills graders reward on the free-response section, where a correct number with a wrong interpretation still loses credit.

Interpreting the interval and the confidence level

Two different sentences are easy to mix up, so keep them separate.

Interpret one interval: you are C% confident that the interval from $a$ to $b$ contains the population mean (or mean difference), stated with the response variable and population in context. For a 95% interval of (113.79, 122.21) grams you would say you are 95% confident the true mean tomato weight is between 113.79 and 122.21 grams.

Interpret the confidence level: in repeated random sampling with the same sample size from the same population, about C% of the intervals built this way would capture the true mean. The level describes the long-run method, not one particular interval. For the exact wording, see what 95 percent confidence means.

Using an interval to justify a claim

A confidence interval gives a range of plausible values for the mean, so it can serve as evidence for or against a claim. If a claimed value falls inside the interval, the data are consistent with that claim and you do not have convincing evidence against it. If the claimed value falls outside the interval, the interval provides convincing evidence against it.

This is why an interval is a natural companion to a two-sided test: a value rejected by a test at significance level $\alpha$ is exactly a value outside the matching $100(1-\alpha)\%$ interval. Every justification should refer to the parameter and the population, not just the numbers.

Sample size, width, and margin of error

For a fixed sample, raising the confidence level increases the critical value $t^*$, which increases the margin of error and widens the interval. More confidence costs precision.

A larger sample size decreases the standard error $s/\sqrt{n}$, so with everything else held the same, the interval tends to get narrower as $n$ grows. The width is roughly proportional to $\dfrac{1}{\sqrt{n}}$, so to halve the margin of error you need about four times the sample size. Confirm the pieces with the margin of error calculator and the confidence interval calculator.