Unknown standard deviation

In the examples on the previous page, the population standard deviation, σ, was a known value. Unfortunately this is rarely the case in practice, so the previous test cannot be used.

Returns from Mutual Funds

This example shows why the previous test statistic cannot be used when σ is an unknown parameter.

Test statistic and its distribution

The previous z-statistic cannot be used in hypothesis tests when the population standard deviation, σ, is an unknown value. Defining the statistical distance from the sample mean to the null-hypothesis value of μ allows it to be used as a test statistic, but it no longer has a standard normal distribution.

Distribution of the t statistic

This simulation demonstrates that the t statistic has greater spread than a standard normal distribution when the sample size is small.

Shape of the t distribution