Provide three examples of studies when we can use (a) the pooled, (b) non-pooled, and (3) paired inference to compare means of two populations.

In a problem of comparing means of two populations we may come across the following conditions depending on the property of data.
(a)Pooled variance: when the data selected for comparing two population variance.
Then we used polled variance to draw interefences and compare the means.
It $s_1^2$ and $s_2^2$ are the sample variances of two population with equal population variances, then "pooled variance" is given by
${\displaystyle s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}$
In this case, the test statistic for comparing means is given by
${\displaystyle A=\frac{\overline{x_1}-\overline{x_2}}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\text{with}(n_1+n_2-2)df}$
(b)Non - pooled variance:
On the contrary the property discused in part(a) about the data, when the populations variances of two populations are assumed to be unequal, then non-pooled variance is used to compare the means of two population.
It is given as
${\displaystyle S\text{non-pooled}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$
and the test statistic is
${\displaystyle A=\frac{\overline{x_1}-\overline{x_2}}{s_p\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}‘}$
(c)Paired interfence(Pained test):
Let us consider the case when
(i) sample a sizes are equal and
(ii) the samples are not independent but the sample observations are painted together.
i.e. the pair of observations${\displaystyle (x,iy.i)(i=1,2..n)}$
comprresponding to the same i-th unit.
For example. Suppose we want to test the effiency of a particular drug, say, for inducing sleep.Let xi and ${\displaystyle yi(i=1..n)}$ be the readings, in hours of sleep, an the i-th individual before and after the drug is givenrespectively.