Give a student response to the following: When testing the hypothesis H0: 1=2 versus Ha: 1=2, the procedure depends on what we are assuming or what we know about the population variances. Here's how the test changes in each scenario:

1. Known and Equal Variances

• Test used: Two sample z-test for the difference of means.
• Assumptions:

And both are known.

Populations are normally distributed (or large n, by CLT)

• Test statistic:

where 2 is the known (equal) population variance.

• Critical values come from the standard normal distribution (z-distribution)

2. Known but Unequal Variances:

• Test used: Two-sample z-test, but with adjusted standard error.
• Assumptions:

and both are known.

• Test statistic:

• Again, use z-distribution for critical values.

3. Unknown but Equal Variances:

• Test used: Two-sample pooled t-test.
• Assumptions:

but unknown.

• Populations are normal.
• Pooled variance estimate:

• Test statistic:

Use the t-distribution with df=n1+n22

4. Unknown and Unequal Variances.

• Test used: Welch's t-test.
• Assumptions:

both are unknown.

• Test statistic:

• Degrees of freedom are approximated using Welch-Satterthwaite equation: Use t-distribution with this approximate df.