Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- p is fixed; intervals vary from sample to sample. A single interval may or may not contain p.
- Interval template: “We are C% confident that the interval from a to b captures the proportion of all [population] who [response].”
- Confidence level template: “If we took many random samples of size n and built a C% interval from each, about C% of the intervals would capture the true proportion of [context].”
- An interval is a set of plausible values for p.
Key relationships
| Situation | Conclusion or effect |
|---|---|
| Claimed value outside the interval | Convincing evidence against p = that value |
| Claimed value inside the interval | Plausible; no convincing evidence against it; not proved |
| Whole interval on one side of a boundary (e.g. all above 0.5) | Convincing evidence for the claim on that side |
| Interval straddles the boundary | Not convincing evidence either way |
| Higher confidence level (same data) | z* up, MOE up, width up; SE unchanged |
| Larger sample (same p̂, same level) | SE down, MOE down, width down |
| n multiplied by k | width divided by about √k (4 × n halves it) |
Assumptions
- The interval only means something if the Topic 3.3 conditions (random, 10%, large counts) hold.
- The margin of error covers random sampling variation only, not bias.
Mistakes to avoid
- “95% probability that p is between a and b.” Use “95% confident”.
- Describing individuals: the interval is not where 95% of people’s values lie.
- Saying a plausible value is proved or “accepted”.
- Leaving out the population in the interpretation.
- Thinking higher confidence means more precision. It means a wider interval.
- Doubling n to halve the margin of error. You need 4 times n.
Quick self-check
- A 95% interval is (0.62, 0.70). Is there convincing evidence against a claim of 0.75? Against 0.65? (0.75: yes, it is outside. 0.65: no, it is plausible, but not proved.)
- The sample size is multiplied by 9. What happens to the margin of error? (It is divided by about 3.)
- 100 random samples each give a 95% interval. About how many capture p? (About 95, but the exact number varies.)
Next: practice questions.