Two independent samples of sizes 20 and 25 are randomly selected from two normal populations with equal variances. In order to test the difference between the population means, the test statistic is: a. approximately standard normal random variable b. a standard normal random variable c. t-distributed with 43 degrees of freedom d. t-distributed with 45 degrees of freedom

The random variable t is distributed , with the degrees of freedom

And the pooled variance can be founded with the following formula:

The degrees of freedom are given by:

So then the best answer for this case:

c. t-distributed with 43 degrees of freedom

Step-by-step explanation:

When we use a a two-sample equal-variance t-test, the basic assumptions are "that the distributions of the two populations are normal, and that the variances of the two distributions are the same".

Both are normally distributed but without the variance equal for both populations.  

The system of hypothesis can be:

Null hypothesis:

Alternative hypothesis:

We can define the following random variable:

The random variable t is distributed , with the degrees of freedom

And the pooled variance can be founded with the following formula:

The degrees of freedom are given by:

c. t-distributed with 43 degrees of freedom

3, 12, 48, 384, 1536, 6144, 24576, 98304

step-by-step explanation:

just keep timing it by 4.