Give a student reply to how well they explained the following: When comparing the means of two approximately normally distributed populations, the hypothesis testing procedure changes depending on the information available about population variances. If the variances are known and equal, we use a two-sample Z-test assuming both populations share the same variance. The test statistic is calculated by subtracting the sample means and dividing by the square root of the common variance times the inverse of each sample size. This method relies on the Z-distribution.

If the variances are known but unequal, we still use a Z-test, but we adjust the formula to account for the different variances from each population. The denominator becomes the square root of the sum of each variance divided by its respective sample size. This test is more conservative, as it doesn't assume a shared population spread.

When the variances are unknown but assumed to be equal, we switch to a pooled two-sample t-test. This requires calculating a pooled sample variance, which is a weighted average of the two sample variances. The test statistic is then computed using this pooled variance, and the degrees of freedom are equal to the total sample size minus two. This method is common in practice but only appropriate when the equal variance assumption is justified.

Finally, if the variances are unknown and unequal, we use Welch's t-test. This test does not assume equal population variances and does not pool the sample variances. Instead, it uses the individual sample variances and sample sizes to compute the test statistic. The degrees of freedom are approximated using the Welch-Satterthwaite equation, which often results in a non-integer value. Welch's t-test is generally more robust, especially when the sample sizes are unequal.