To estimate the road test failure rate, a random sample of 65 student drivers is used, among whom 37 failed at least once. Which interval method should be used?

To estimate the road test failure rate, a random sample of 65 student drivers is used, among whom 37 failed at least once. Which interval method should be used?

• A. 2-proportion z-interval
• B. Paired t-interval
• C. 1-proportion z-interval
• D. z-interval for mean

Answer: (C)

Estimating a population proportion from a single sample. Here we want the proportion of all student drivers who fail the road test at least once. With 65 students and 37 failures, the sample proportion is p̂ = 37/65 ≈ 0.569. For a single proportion, the standard approach is a 1-proportion z-interval, which uses the normal approximation: p̂ ± z* sqrt[p̂(1−p̂)/n]. The large-sample condition is satisfied here (np̂ ≈ 37 and n(1−p̂) ≈ 28), so the normal approximation is reasonable. That’s why this method is the best choice. Other interval methods serve different questions: a 2-proportion interval compares two independent proportions, a paired t-interval applies to differences in means from paired data, and a z-interval for mean estimates a population mean rather than a proportion.

Estimating a population proportion from a single sample. Here we want the proportion of all student drivers who fail the road test at least once. With 65 students and 37 failures, the sample proportion is p̂ = 37/65 ≈ 0.569. For a single proportion, the standard approach is a 1-proportion z-interval, which uses the normal approximation: p̂ ± z* sqrt[p̂(1−p̂)/n]. The large-sample condition is satisfied here (np̂ ≈ 37 and n(1−p̂) ≈ 28), so the normal approximation is reasonable. That’s why this method is the best choice. Other interval methods serve different questions: a 2-proportion interval compares two independent proportions, a paired t-interval applies to differences in means from paired data, and a z-interval for mean estimates a population mean rather than a proportion.