Question # 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.

Comparing two groups
ANSWERED 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. 2020-12-26

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
$$\displaystyle{{s}_{{{p}}}^{{{2}}}}\ =\ {\frac{{{\left({n}_{{{1}}}\ -\ {1}\right)}\ {{s}_{{{1}}}^{{{2}}}}\ +\ {\left({n}_{{{2}}}\ -\ {1}\right)}\ {{s}_{{{2}}}^{{{2}}}}}}{{{n}_{{{1}}}\ +\ {n}_{{{2}}}\ -\ {2}}}}$$
In this case, the test statistic for comparing means is given by
$$\displaystyle{A}\ =\ {\frac{{\overline{{{x}_{{{1}}}}}-\overline{{{x}_{{{2}}}}}}}{{{s}_{{{p}}}\sqrt{{{\frac{{{1}}}{{{n}_{{1}}}}}+{\frac{{{1}}}{{{n}_{{2}}}}}}}}}}\ \text{with}\ {\left({n}_{{1}}\ +\ {n}_{{2}}\ -\ {2}\right)}\ {d}{f}$$
(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$$\displaystyle{\left({x},{i}{y}.{i}\right)}\ {\left({i}={1},{2}..{n}\right)}$$
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 $$\displaystyle{y}{i}\ {\left({i}\ =\ {1}..{n}\right)}$$ be the readings, in hours of sleep, an the i-th individual before and after the drug is givenrespectively.