Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- μ (or μd) is fixed; x̄ and s change from sample to sample, so both the centre and the width of the interval change. One interval may or may not capture the parameter.
- Interval template: “We are C% confident that the interval from a to b [units] captures the true mean [variable] of all [population].”
- Matched pairs: “…captures the true mean difference in [variable] (first − second) for [population].”
- Confidence level template: “If we took many random samples of size n and built a C% t-interval from each, about C% of the intervals would capture the true mean [context].”
- An interval is a set of plausible values for the parameter.
Key relationships
| Situation | Conclusion or effect |
|---|---|
| Claimed value outside the interval | Convincing evidence against the parameter equalling that value |
| Claimed value inside the interval | Plausible; no convincing evidence against it; not proved |
| Interval for μd entirely above (or below) 0 | Convincing evidence of a positive (or negative) mean difference |
| Interval for μd contains 0 | No convincing evidence of a difference |
| Higher confidence level (same data) | t* up, MOE up, width up; SE = s/√n unchanged |
| Larger sample (same x̄, s, level) | SE down, t* slightly down, MOE and width down |
| n multiplied by 4 | width roughly halved (slightly less than half for t) |
Assumptions
- The Topic 4.2 conditions hold: random sample or random assignment; n ≤ 10% of the population when sampling without replacement; normal population, n ≥ 30, or no strong skew or outliers in the sample (of differences, for pairs).
- A cause-and-effect conclusion needs random assignment; generalising to a population needs random sampling.
- The margin of error covers random variation only, not bias.
Mistakes to avoid
- “95% probability that μ is between a and b.” Use “95% confident”.
- Describing individuals: the interval is about the mean, not where 95% of values lie.
- Leaving out the order of subtraction for μd.
- “0 is inside, so there is no difference.” Say there is no convincing evidence of a difference.
- Interpreting about x̄ or the sample. The interval always contains x̄.
- Thinking higher confidence means more precision. It means a wider interval.
Quick self-check
- A 95% interval for a mean is (12.4, 15.0) cm. Is there convincing evidence against a claim of 15.5 cm? Against 13 cm? (15.5: yes, it is outside. 13: no, it is plausible, but not proved.)
- With d = after − before, a 90% interval for μd is (−2.5, −0.8) points. What does it suggest? (Every plausible value is negative, so there is convincing evidence that the mean score fell from before to after.)
- The sample size is multiplied by 9 with the same x̄ and s. What happens to the width? (It is divided by about 3, slightly more because t also falls.)*
Next: practice questions.