When we replace the population standard deviation σ with the sample standard deviation (s) in our standardized test statistic formula, the distribution of the test statistic changes to a t-distribution.

This occurs because the use of the sample standard deviation introduces additional variability, especially for smaller sample sizes, and the t-distribution accounts for this variability with its heavier tails compared to the normal distribution.

Would you like a detailed explanation of how this impacts hypothesis testing?

Here are 5 related questions for further exploration:

1. What are the key differences between the normal distribution and the t-distribution?
2. How does the shape of the t-distribution change as the degrees of freedom increase?
3. Why is the t-distribution necessary for smaller sample sizes?
4. How do we calculate the degrees of freedom for a t-test?
5. In what situations can we approximate the t-distribution to a normal distribution?

Tip: Always ensure that your sample size is sufficiently large if you wish to approximate the t-distribution with a normal distribution.