A confidence interval gives the plausible values of a parameter. If a claimed value of μ (or μd) is inside the interval, the data do not give convincing evidence against the claim. If it is outside, they do. Set up an interval, move the claim, and practice writing the justification.
Pick a preset or type your own summary statistics. Then move the claimed value.
For the current s, each curve shows the margin of error at every n. The dot is your interval. A claim just outside a narrow interval might be inside a wider one.
For each 95% interval, decide what it says about the claim.
Every value inside the confidence interval is a plausible value of the parameter, based on the data. Values outside the interval are not plausible at that confidence level.
If the claimed value is in the interval, the data do not give convincing evidence that the claim is wrong. That is not proof the claim is right: many other values are plausible too.
If the whole interval is below the claim, the data give convincing evidence that the true mean is less than the claim. If it is all above, convincing evidence that it is greater.
For paired data the usual claim is μd = 0 (no difference on average). An interval entirely above 0 or entirely below 0 is convincing evidence of a real mean difference in that direction.
A C% interval and a two-sided test at α = 1 − C agree: the claim is rejected by the test exactly when it falls outside the interval. A 95% interval matches a two-sided test at α = 0.05.
Larger samples give narrower intervals, which can rule out claims a small sample could not. Higher confidence gives wider intervals, which rule out fewer claims.
Sentence starters for justifying a claim with an interval. Always include the interval, the claim, and context.
| Where is the claim? | Is it plausible? | What to write |
|---|---|---|
| Inside the interval | Yes | "Because 16 is in the interval (15.7, 16.2), 16 oz is a plausible value for the true mean weight. We do not have convincing evidence that the mean differs from 16 oz." |
| Below the whole interval | No | "Because every value in (101.2, 108.9) is above 100, the data give convincing evidence that the true mean is greater than 100." |
| Above the whole interval | No | "Because all plausible values are below 16, there is convincing evidence that the true mean fill is less than 16 oz." |
| Paired, interval contains 0 | 0 is plausible | "Since 0 is in the interval, it is plausible there is no mean difference. The data are not convincing evidence that the program changes scores." |
| Paired, interval all above 0 | 0 is not plausible | "All values in the interval for μafter − before are positive, so there is convincing evidence of a mean increase." |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 4.3. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.