Why is it necessary to assume equal variances in two sample t tests?

[Faiq]

Homework Statement

My question is why is the assumption necessary to make? (Please see the image).

Homework Equations

The Attempt at a Solution

We can easily proceed by treating the two samples as two different population, find their individual unbiased estimate of variance and then use the combined estimate to apply the Central Limit Theroem. So why assume the variances of both samples is equal?

 

[Ray Vickson]

Homework Statement

My question is why is the assumption necessary to make? (Please see the image).

Homework Equations

The Attempt at a Solution

We can easily proceed by treating the two samples as two different population, find their individual unbiased estimate of variance and then use the combined estimate to apply the Central Limit Theroem. So why assume the variances of both samples is equal?

The Central Limit Theorem applies in the limit of infinite sample slzes and gives a reasonable approximation for very large but finite samples. However, for sample sizes like in the question you cite, the central limit theorem does not apply very well, if at all, so you need to use the t-distribution instead of the normal. For a two-sample t-test you need equality of variances just because without it the test statistic does not follow the t-distribution. It is just a requirement of the mathematics.

 

[Faiq]

Oh so equality of variances is necessary for any two sample t or z test ?

 

[Ray Vickson]

Oh so equality of variances is necessary for any two sample t or z test ?

Yes, that is what I said.

However, in a paired-sample t-test you do not need equality of variances, so be careful to make the distinction between"two-sample" and "paired-sample" (both of which look at two samples, but in different ways).

 

[Faiq]

Thank you very much

 

[Faiq]

Can you please tell me what test am I suppose to use in these situations (2 sample z test or 2 sample t test)?
1. Two sample, sample size small, variance of both known and is different
2. Two sample, sample size large, variance of both unknown

I am asking because you said when sample size is small, we have to use t-test for which variance must be equal. In that case I can't identify the type of test to be used in the abovementioned conditions.