To estimate a population mean with a small sample and unknown standard deviation, which method should be used?

To estimate a population mean with a small sample and unknown standard deviation, which method should be used?

• A. 2-sample t-test
• B. 1-sample t-interval
• C. 2-proportion z-test
• D. Paired t-test

Answer: (B)

When you want to estimate a population mean from a small sample and you don’t know the population standard deviation, use a one-sample t-interval. The reason is that you must replace the unknown sigma with the sample standard deviation s, and the t-distribution properly accounts for the extra uncertainty from this estimation. The interval is x̄ ± t_{n-1, α/2} · (s/√n), where n is the sample size and df = n−1. This approach is appropriate specifically for estimating a mean, not for testing or estimating other quantities. Two-sample t-test is for comparing the means of two independent groups, not for estimating a single mean. The paired t-test is for paired or matched measurements on the same units. The 2-proportion z-test is for proportions, not means, and typically relies on large-sample z approximations.

When you want to estimate a population mean from a small sample and you don’t know the population standard deviation, use a one-sample t-interval. The reason is that you must replace the unknown sigma with the sample standard deviation s, and the t-distribution properly accounts for the extra uncertainty from this estimation. The interval is x̄ ± t_{n-1, α/2} · (s/√n), where n is the sample size and df = n−1. This approach is appropriate specifically for estimating a mean, not for testing or estimating other quantities.

Two-sample t-test is for comparing the means of two independent groups, not for estimating a single mean. The paired t-test is for paired or matched measurements on the same units. The 2-proportion z-test is for proportions, not means, and typically relies on large-sample z approximations.