Question

Under what conditions must we assume a Student t distribution for the sampling distribution of sample means when testing a claim about a population mean?

a) We assume a Student t distribution when the population size is large.
b) We assume a Student t distribution when the population standard deviation is known.
c) We assume a Student t distribution when the population standard deviation is unknown, and the sample size is small.
d) We assume a Student t distribution when the population mean is known.

Answer 1

A Student t distribution should be assumed when the population standard deviation is unknown, and the sample size is small, making option c) the correct choice for the conditions under which a Student t distribution is used.

Step-by-step explanation:

When testing a claim about a population mean, we must assume a Student t distribution for the sampling distribution of sample means under certain conditions. Specifically, we assume a Student's t-distribution when the population standard deviation is unknown, and the sample size is small. This is typically taken as a sample size smaller than 30. When you have a small sample size, using the sample standard deviation (s) to estimate the unknown population standard deviation (σ) is more appropriate with the t-distribution. If the sample is a simple random sample and is from a population that is approximately normally distributed or the sample size is sufficiently large, the t-test is considered appropriate.