a) Assumptions large sample inferences about the population difference between population means using two sample z-tests. We have three conditions as follows:

1) The two samples are independent.

2) Two samples are large samples that is \(\displaystyle{n}_{{1}}\ne{q}{30},{n}_{{2}}\ne{q}{30}\) Here, \(\displaystyle{n}_{{1}},{\quad\text{and}\quad}{n}_{{2}}\) are the sizes of the two samples.

b) Assumptions small sample inferences about the population difference between populations means using independent sample design We have three conditions as follows

1) Independent random samples

2) Both populations are normal

3) Both populations are having equal variances, that is \(\displaystyle{\sigma_{{{1}}}^{{{2}}}}={\sigma_{{{2}}}^{{{2}}}}\) Assumptions small sample inferences about the population difference between population means using two sample t-tests.

c) Assumptions required to small sample inferences about \(\displaystyle{\left(\mu_{{1}}-\mu_{{2}}\right)}\) using a paired difference design as follows:

1) Random sample of paired differences

2) Populations of difference are normal Here, \(\displaystyle{n}_{{d}}\) is the number of paired differences considered.

d) Assumptions required to large sample inferences about the differences \((p_1 - p_2)\) using two sample z-tests as follows:

1) Independent random samples

2) \(n_{1}p_{1}\Uparrow\neq 15\), and \(n_{1}q_{1}\Uparrow\neq 15\)

3) \(n_{2}p_{2}\Uparrow\neq 15\), and \(n_{2}q_{2}\Uparrow\neq 15\Uparrow\)

e) Assumptions required to inferences about the ratio \(\displaystyle{\frac{{{\sigma_{{{1}}}^{{{2}}}}}}{{{\sigma_{{{2}}}^{{{2}}}}}}}\) of two population variances using F test. The assumptions are same for either small or large samples.

1) Independent samples.

2) Both populations normal