1. Discuss the basic strategy for comparing the means of two populations based on independent simple random samples.

The basic strategy for comparing the means of two populations based on independent simple random samples is given in the following steps:

1. Independently and randomly take a sample from Population 1 and a sample from Population 2

2. Compute the mean $\overline{x_1}$of the sample from Population 1 and the mean $\overline{x_2}$of the sample from Population 2

3. Reject the null hypothesis if the sample means$\overline{x_1}$ and $\overline{x_2}$differ by too much; otherwise, do not reject the null hypothesis.

Suppose that x is a normally distributed variable on each of two populations. Then, for independent samples of sizes $n_1$and $n_2$from the two populations,

$\text{-}{\mu }_{{\overline{x}}_1-{\overline{x}}_2}={\mu }_1-{\mu }_2$

width="219" height="37" role="math">σx¯1-x¯2=σ12/n1+σ22/n2,and

${\overline{x}}_1-{\overline{x}}_2\text{is normally distributed.}$

Test Statistic:

i. For comparing two population means when the population standard deviations are known.

$z=\frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{\left({\sigma }_1^2/n_1\right)+\left({\sigma }_2^2/n_2\right)}}$

ii. For comparing two population means when the population standard deviations are unknown and assumed to be equal.

$t=\frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{s_{\mathrm{p}}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}~t_{\left(n_1+n_2-2\right)}$

$s_{\mathrm{p}}=\sqrt{\frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}}$

iii. For comparing two population means when the population standard deviations are unknown and assumed to be unequal.

$t=\frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}~t\left(\Delta \right)$

$\text{where}\Delta =\frac{{\left.\left(s_1^2/n_1\right)+\left(s_2^2/n_2\right)\right]}^2}{\frac{{\left(s_1^2/n_1\right)}^2}{n_1-1}+\frac{{\left(s_2^2/n_2\right)}^2}{n_2-1}}$