Topic Review — Interval Estimates for Proportions

← Interval Estimates for Proportions

This review covers all lessons in this topic: confidence intervals for proportions, margin of error and sample size, and applications.

Review Questions

1. Define the margin of error in the context of a confidence interval for a proportion.
2. A sample of 300 people finds 192 prefer online shopping. Construct a 95% confidence interval for the true proportion.
3. Explain the correct interpretation of a 95% confidence interval. What does 95% refer to?
4. State the three critical values z* and their associated confidence levels.
5. A 95% CI for p is (0.33, 0.49). Find the sample proportion p̂, the margin of error, and back-calculate the approximate sample size.
6. Check whether the normality conditions are met for n = 80 and p̂ = 0.06.
7. A researcher wants ME ≤ 0.04 with 95% confidence. No prior estimate of p is available. Find the minimum sample size.
8. A manufacturer says its product has a 15% return rate. A random sample of 200 purchases finds 24 returns. Construct a 95% CI. Is the manufacturer’s claim consistent with the data?
9. Explain why using p̂ = 0.5 in the sample size formula gives the most conservative (largest) estimate of n.
10. A pilot study estimates p̂ ≈ 0.72. A researcher wants a 99% CI with ME ≤ 0.03. Find the required sample size.
11. Two schools are surveyed about homework preferences. School 1: n = 150, 87 prefer less homework. School 2: n = 150, 72 prefer less homework. Construct 95% CIs for each and determine whether there is evidence of a genuine difference.
12. A current 95% CI uses n = 500. How large a sample is needed to reduce the margin of error by one third (to 2/3 of its current value)?
13. A survey of 600 Australians finds 420 support a new environmental policy.
    • (a) Construct a 90% CI.
    • (b) Construct a 99% CI.
    • (c) Compare the widths. What trade-off does this illustrate?
14. An online news poll of 2000 respondents reports 58% support for a proposition with ME = ±2.2%. A statistician says the poll is unreliable. Give one reason why.
15. A 95% CI based on p̂ = 0.48 and n = 400 is constructed. Without recalculating, what would happen to this interval if the confidence level were changed to 90%? To 99%? Describe the effect on both the margin of error and the interval width.