A reliable method
Calculate the margin of error, then subtract and add it to the estimate. For a proportion, check random sampling, independence and adequate success/failure counts for the chosen procedure.
The common trap
A 95% confidence procedure captures the fixed population parameter in about 95% of intervals over repeated valid samples. It does not contain 95% of individual data values.
Make the connection
A higher confidence level generally widens an interval. Larger samples narrow intervals when other conditions remain similar. Neither choice corrects a biased sampling plan.
Endpoint, width and margin
An estimate .60 and SE .02 with multiplier 2 give margin .04 and interval [.56,.64]. The full width is .08. Reporting .04 as an endpoint omits the estimate; reporting .08 as the margin doubles the intended uncertainty. Draw the estimate in the middle and label each half-width.
Interpret repeated sampling carefully
A 95% method aims for long-run coverage of the fixed parameter. It does not promise that 95% of observations lie between the endpoints, nor that the parameter moves from interval to interval. Wider intervals from higher confidence do not correct nonresponse or measurement bias.
Transfer check
An observed interval for a mean is [21,23]. The estimate is 22 and margin is 1. Explain what would happen to the margin if the sample size quadrupled, assuming similar variability and the same critical multiplier. State the independence and sampling assumptions that make the comparison reasonable.
A proportion estimate is 1/2 with SE = 1/50. Using multiplier 2, what is the upper interval endpoint?
- Margin of error = 2 × 1/50 = 1/25.
- Add the margin to the estimate.
- Upper endpoint = 27/50.
Compare the other answer choices
Use one standard error and omit the critical multiplier.
Report a signed margin, omitting the estimate.
Use the full interval width as the margin.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Statistics 2026–27 course framework