Question

How does a t-distribution differ from the z-distribution? How do degrees of freedom (df) affect this? What is the effect of this difference on hypothesis testing?

- Your friend performed a two-tailed experiment in which n = 20. He couldn’t find his t-table, but remembered the t-critical at df = 10. He decided to compare his t-obtained to this t-critical and determined the results were not significant. Is this ok? Why or why not.

Answer 1

A z distribution is used when the population standard deviation is known. A t distribution is used when the population standard deviation is not known. When the population standard deviation is not known, the sample standard deviation is used instead, and the t distribution is used to perform the hypothesis testing.

Now, the degrees of freedom are equal to the sample size (n) minus 1. Thus, if the sample size is 10, the degrees of freedom are 9.

As the degrees of freedom increase, the t distribution begins to increasingly look like the z distribution. When the degrees of freedom are very large, the z distribution and the t distribution are exactly the same.

Thus, when we use a z distribution, the confidence interval at a certain significance level is narrower than the confidence interval for the same significance level of a t distribution. As the degrees of freedom keep increasing, the size of the intervals keeps getting similar.

What the friend has decided to do is not okay. This is because the t values at a degree of freedom of 10 are larger than the t values at a degree of freedom of 20. Thus, the confidence interval obtained at a degree of freedom of 10 is wider than the interval obtained at a degree of freedom of 20. Thus, the results which were not significant at 10 may be significant at 20.

Thus, what the friend has decided to do is not okay.