Forty people with allergies take an antihistamine; they report that their discomfort subsided on average in 18 minutes with a standard deviation of 4 minutes. The goal is to estimate the population relief time. Which method is appropriate?

Forty people with allergies take an antihistamine; they report that their discomfort subsided on average in 18 minutes with a standard deviation of 4 minutes. The goal is to estimate the population relief time. Which method is appropriate?

• A. 1-Sample T-Interval
• B. 2-Sample T-Interval
• C. 1-Proportion Z-Interval
• D. Chi-Square Test

Answer: (A)

When you want to estimate a population mean from a single sample of quantitative data and you don’t know the population standard deviation, use a 1-sample t-interval. This fits here because there’s one group, with a mean relief time, and the population SD is unknown, so we rely on the sample standard deviation and the t distribution. You compute the interval with x̄ ± t*(s/√n), where df = n−1. Here n is 40, x̄ is 18, and s is 4. The standard error is 4/√40 ≈ 0.632. For a typical 95% confidence level, the t critical value with 39 degrees of freedom is about 2.02, giving a margin of error ≈ 2.02 × 0.632 ≈ 1.28. So the 95% confidence interval for the population mean relief time is roughly 18 ± 1.28, i.e., about 16.7 to 19.3 minutes. The other options don’t fit: a two-sample t-interval compares means from two groups, not a single sample. a 1-proportion interval estimates a population proportion, not a mean. a chi-square test is for variance or categorical data, not for estimating a mean.

When you want to estimate a population mean from a single sample of quantitative data and you don’t know the population standard deviation, use a 1-sample t-interval. This fits here because there’s one group, with a mean relief time, and the population SD is unknown, so we rely on the sample standard deviation and the t distribution.

You compute the interval with x̄ ± t*(s/√n), where df = n−1. Here n is 40, x̄ is 18, and s is 4. The standard error is 4/√40 ≈ 0.632. For a typical 95% confidence level, the t critical value with 39 degrees of freedom is about 2.02, giving a margin of error ≈ 2.02 × 0.632 ≈ 1.28. So the 95% confidence interval for the population mean relief time is roughly 18 ± 1.28, i.e., about 16.7 to 19.3 minutes.

The other options don’t fit: a two-sample t-interval compares means from two groups, not a single sample. a 1-proportion interval estimates a population proportion, not a mean. a chi-square test is for variance or categorical data, not for estimating a mean.