I'm not sure what you're asking for.

A t-distribution is used when we need to replace the population variance

with the sample varaince, in our test statistic.

If we have normality, the distribution is now a t with n-1 degrees of freedom. As n goes towards infinity the t converges to a standard normal.

But you can see that from the definition of a t.

It's a N(0,1) divided by the square root of a chi-square divided by it's degrees of freedom. That denominator will go to one as n goes to infinity.

SO, basically do what you did in the Z case, but use the percentile points from a t distribution.

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There's some decent stuff at

Student's t-test - Wikipedia, the free encyclopedia

But it's a little weird there.

I don't know why they even mention the equal variance and equal sample size in the two sample case.

The pooled sample variance is a weighted average of the two sample variances.

Clearly, when the two sample sizes are equal the pooled sample variance

is obviously one half of the sum of the two sample variances.

If you have a specific question let me know.