The one-sample t-test is the appropriate test for a claim about a population mean when
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → the population is roughly normal, its standard deviation is unknown, and the sample is small
The t-test is a test for the mean of a population that is normal or roughly normal and whose standard deviation is unknown. The standard deviation is then estimated by s from a small sample, and t is found from \( t = \displaystyle\frac{\bar{x} - \mu}{s/\sqrt{n}} \).
If \(\mu\) were already known from the whole population, there would be nothing to test. A hypothesis is always about a population value such as \(\mu\), never about \(\bar{x}\), which is known once the sample is taken. A known standard deviation with a very large sample is not the situation the t-test is meant for.