Rarely does a researcher wishing to test a hypothesis about a population's mean already know the population's standard deviation (σ). Therefore, testing hypotheses about means almost always involves the estimation σ. When σ is known, the formula:

is used for a confidence interval. When σ is not known,

is used as an estimate of σ

Suppose a researcher wished to test whether the mean score of fifth graders on a test of reading achievement in his or her city differed from the national mean of 76. The researcher randomly sampled the scores of 20 students. The scores are shown below.

72 69 98 87 78 76 78 66 85 97 84 86 88 76 79 82 82 91 69 74