What is the single-sample t test used for?

• A. A sample mean to a known population mean when the population standard deviation is unknown.
• B. A sample mean to a known population mean when the population standard deviation is known.
• C. Two related sample means.
• D. Two independent sample means.

Answer: (B)

The main idea is to compare a single sample’s average to a known population average when you don’t know the population’s variability. Because the population standard deviation is unknown, you estimate it with the sample standard deviation, and the test uses the t distribution to reflect this extra uncertainty. The result is a t statistic calculated as (sample mean minus hypothesized mean) divided by (sample standard deviation divided by the square root of the sample size). If this statistic is large in magnitude and the p-value is small, you conclude the sample mean differs from the hypothesized population mean. If the population standard deviation were known, you’d use a z test instead. The other tests involve comparing two means rather than a single mean to a known value: a paired (related) design and an independent (unrelated) design.