Evaluating a new drug. Merck Research Labs experimented to evaluate the effect of a new drug using the single-T swim maze. Nineteen impregnated dam rats were captured and allocated a dosage of 12.5 milligrams of the drug. One male and one female rat pup were randomly selected from each resulting litter to perform in the swim maze. Each pup was placed in the water at one end of the maze and allowed to swim until it escaped at the opposite end. If the pup failed to escape after a certain period, it was placed at the beginning of the maze and given another chance. The experiment was repeated until each pup accomplished three successful escapes. The table below reports the number of swims required by each pup to perform three successful escapes. Is there sufficient evidence of a difference between the mean number of swims required by male and female pups? Conduct the test (at a = .10). Comment on the assumptions required for the test to be valid.

Source: Copyright © 2012 by Merck Research Laboratories.

$H_0:$There is no statistically significant difference in the average number of swims needed by the male and female pups to complete three main escapes.

$Ho:\left({\mu }_1-{\mu }_2\right)=0$

$H_{\alpha }:$ The average amount of swims needed by the male and female paps to complete three effective escapes differs significantly.

Under $H_0$test statistics:

$$\begin{gathered} t=\frac{\left({\overline{x}}_1-{\overline{x}}_2\right)}{\sqrt{{\left(\frac{S_1}{n_1}\right)}^2+{\left(\frac{S_2}{n_2}\right)}^2}} \\ =\frac{\left(5.8947-5.5263\right)}{{\left(\frac{\sqrt{2.378}}{19}\right)}^2+{\left(\frac{\sqrt{2.4578}}{19}\right)}^2} \end{gathered}$$

$$\begin{gathered} =\frac{0.3684}{\sqrt{{\left(0.125\right)}^2+{\left(0.129\right)}^2}} \\ =\frac{0.3684}{\sqrt{0.015625+0.016641}} \\ =\frac{0.3684}{0.032266} \\ =0.4696 \end{gathered}$$

Rejection regions are$t>t_{\alpha /2}\text{or}t<-t_{\alpha /2}$ where the tale is dependent on $\left(n_d-1\right)$degrees of freedom. Degrees of freedom at a level of 0.10.

Because our test statistic is in the acceptance range, one cannot reject the null hypothesis and may infer that there is a 10% difference in the mean number of Swims necessary to complete four successful escapes by male or female pups.

Conditions required for valid small sample inferences about$\left({\mu }_1-{\mu }_2\right)$

1. The two samples are drawn at random from the two target populations in an independent way.
2. The distributions of the sampled groups are roughly normal.
3. The variations in the population are equivalent.