$8.-$ The Bureau of Labor Statistics publishes data on weekly earnings of full$-$time wage and salary worke

in Employment and Earnings. Independent random samples of male and female workers gave the

following data on weekly earnings, in dollars.

The Normal probability plots and summary statistics are as follows.

a$)$ First do the test to find out if ${}_1$ and ${}_2$ are equal or not. Define $=0\mathrm{.}10$

i$)$ The test statistic value is $($inside$)($outside$)$ the Rejection Region.

ii$)$ The $p-$value is:

b$)$ Test the null hypothesis $H_0$:${}_1={}_2$ versus the alternative $H_1$:${}_1{}_2.$ Define $=0\mathrm{.}05$

i$)$ The test statistic value is $($inside$)($outside$)$ the Rejection Region.

ii$)$ The p$-$value is:

c$)$ In the following graph identify the Type I Error area, $,$ and the Type II Error area, $.$ The red

curve is the null hypothesis probability curve $({}_1-{}_2=0),$ and the blue curve is one of the

alternative hypothesis probability curve $({}_1-{}_2=-2).$

$($In this graph the difference $-2$ is easier to plot than other values corresponding to these data. What is important is the concept$)$

$"$: $($please circle your answer$)$

The green area is $,$ the

probability of Type I Error. The

other area, gold, is $.$

The green area is $,$ the

probability of Type II Error. The

other area, gold, is $$