Weekly Earnings. The Bureau of Labor Statistics published data on weekly earnings of full-time wage and salary workers in Employment and earnings. Male and female workers were paired according to occupation and experience. Their weekly earnings, is dollars, are provided on the WeissStats site. Use the technology of your choice to do the following.

a. Apply the paired $t$test to decide, at the $5\%$significance level, whether the data provide sufficient evidence to conclude that, on average, the weekly earnings of male full-time wage and salary workers exceed those of women.

b. Find and interpret a $90\%$confidence interval for the difference between the mean weekly earnings of male and female full-time wage and salary workers. Use the paired $t$-interval procedure.

c. Obtain a normal probability plot, boxplot, and stem-and-leaf diagram of the paired differences.

d. Based on your results in part (c), are your procedures in parts (a) and (b) justified? Explain your answer.

Given in the question that, the Bureau of Labor Statistics published data on weekly earnings of full-time wage and salary workers in Employment and earnings.

We must determine whether male full-time wage and salary workers earn more than women on a weekly basis.

According to the information,

$\alpha =5\%$

$c=90\%$

We must compute the sample mean using the difference values:

role="math" localid="1653196629861" $$\begin{gathered} \overline{d}=\frac{87+172+\dots +177+166}{50} \\ \approx 131.08 \end{gathered}$$

Compute the standard deviation for the difference in values:

role="math" localid="1653196640184" $$\begin{gathered} s_d=\sqrt{\frac{(87-131.08{)}^2+\dots +(131.08{)}^2}{50-1}} \\ \approx 93.7576 \end{gathered}$$

It is assumed that the female mean is lower than the male mean.

This statement could be either a null or alternative hypothesis.

$H_0:{\mu }_d=0$

$H_{\alpha }:{\mu }_d>0$

The value of the test statistic must now be calculated.

$$\begin{gathered} t=\frac{\overline{d}-{\mu }_d}{\frac{s_d}{\sqrt{n}}} \\ =\frac{131.08-0}{\frac{93.7576}{\sqrt{\overline{x}}}} \\ \approx 9.886 \end{gathered}$$

Let's find the degree of freedom:

$$\begin{gathered} df=n-1 \\ =50-1 \\ =49 \end{gathered}$$

The null hypothesis will be rejected if the p value is lower than the significance level.

Given in the question that,

$c=90\%$

$\alpha =5\%$

For the difference in the mean weekly earnings of male and female full-time wage and salary workers, we must calculate and interpret a 90% confidence interval. Make use of the paired t-interval method.

From the previous exercise we know that, the degree of freedom is : $49$

Now, We have to compute the critical value:

$$\begin{gathered} \alpha =\frac{1-c}{2} \\ =0.05 \end{gathered}$$

$t^{*}=1.677$

So, the margin of error will be:

$$\begin{gathered} E=t^{*}\times \frac{s_d}{\sqrt{n}} \\ =1.677\times \frac{93.7576}{\sqrt{50}} \\ \approx 22.2359 \end{gathered}$$

For $u_d$,the confidence level endpoints can be computed

$$\begin{gathered} \overline{d}-E=131.08-22.2359 \\ =108.8441 \end{gathered}$$

$$\begin{gathered} \overline{d}+E=131.08+22.2359 \\ =153.3159 \end{gathered}$$

Given in the question that, Male and female workers were paired according to occupation and experience.

The paired differences must be shown by a normal probability plot, a boxplot, and a stem-and-leaf diagram.

The normal probability chart is a graphic tool for determining if a data set is roughly normally distributed. The data is shown against with a theoretical normal distribution with the points forming an approximate straight line.

Here, the normal probability plot will be:

A box plot depicts statistical data using the minimum, first quartile, median, third quartile, and maximum values.

To begin, all values should be rounded to the nearest tenth:

$$\begin{gathered} -80,-60,-30,20,20,30,30,30,40,50,60,70,70,70,90,90,90,100,100, \\ 100,100,110,110,120,120,130,150,150,150,160,160,160,170,170,170, \\ 170,17-180,190,200,210,210,210,220,250,270,280,310,350,360. \end{gathered}$$

Now, using a vertical line, position the digits of the hundreds to the left and the digits of the tens to the right as follows:

We must decide whether or not procedures in parts (a) and (b) may be justified based on the outcomes in part (c).

Part (c) of the Box-Plot shows that there are three outliers that greatly influence the $t$-procedure, hence the results in parts (a) and (b) cannot be supported.