How to derive p-value from standard error between two group means?

Question

Data was collected on binary variable and continuous variable. The binary variable represents two groups.

• Group 1 has a sample size $n_1$ and mean $\mu_1$ on the continuous variable
• Group 2 has a sample size $n_2$ and mean $\mu_2$ on the continuous variable

The significance test is as follows:

• H0: μ1 = μ2 versus Ha: μ1 > μ2, using an $\alpha$ significance level.

The first sample mean was $z$ pooled standard errors above the second sample mean.

• How can the p-value be derived from the standard error $z$?
• What if $n_1=15$, $n_2=30$, $\alpha=.10$ and $z=1.6$, what would the p-value be?

Answer 1

From what you read in the question, you know something about the difference between the sample means.

Now if you could only find out what the distribution of that difference in sample means is (under the assumptions that the true difference is zero)?

"With a little luck" then you can find out where the difference you observed can be found in that distribution, which will lead directly to the p-value.