A factory wants to estimate the average weight of its bottles. A random sample of 10 bottles is weighed. Which method should be used to estimate the population mean weight?

A factory wants to estimate the average weight of its bottles. A random sample of 10 bottles is weighed. Which method should be used to estimate the population mean weight?

• A. 2-sample t-interval
• B. 1-sample t-interval
• C. Z-interval
• D. Paired t-interval

Answer: (B)

Estimating a single population mean from a single random sample when the population standard deviation is unknown requires using the one-sample t-interval. Here you’ve got a small sample (10 bottles) and you don’t know the true variability of bottle weights, so you rely on the sample standard deviation and the t-distribution with degrees of freedom n−1. This interval, centered on the sample mean, broadens a bit to reflect the extra uncertainty compared to using a known sigma, giving you x-bar ± t*(s/√n). The other options don’t fit this situation: a two-sample t-interval is for comparing two separate groups, not for a single mean. A Z-interval would be appropriate only if the population standard deviation were known or if the sample were large enough to justify using the normal distribution, which isn’t the case here. A paired t-interval is for matched or before-after data on the same subjects, which isn’t the scenario described.

Estimating a single population mean from a single random sample when the population standard deviation is unknown requires using the one-sample t-interval. Here you’ve got a small sample (10 bottles) and you don’t know the true variability of bottle weights, so you rely on the sample standard deviation and the t-distribution with degrees of freedom n−1. This interval, centered on the sample mean, broadens a bit to reflect the extra uncertainty compared to using a known sigma, giving you x-bar ± t*(s/√n).

The other options don’t fit this situation: a two-sample t-interval is for comparing two separate groups, not for a single mean. A Z-interval would be appropriate only if the population standard deviation were known or if the sample were large enough to justify using the normal distribution, which isn’t the case here. A paired t-interval is for matched or before-after data on the same subjects, which isn’t the scenario described.