In testing a hypothesis about a population mean with an unknown population standard deviation ($\sigma$ )the degrees of freedom is used in the denominator of the test statistic.

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When testing a hypothesis about a population mean with an unknown population standard deviation, the sample standard deviation is used to estimate the population standard deviation, and the degrees of freedom (df) are used to determine the critical values from the t-distribution, not in the denominator of the test statistic. The test statistic itself is calculated using the sample mean, hypothesized population mean, sample standard deviation, and sample size.

In Testing a Hypothesis About a Population Mean with an Unknown