Question

Using Nonparametric Tests. In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.

Job Stress and Income Listed below are job stress scores and median annual salaries (thousands of dollars) for various jobs, including firefighters, airline pilots, police officers, and university professors (based on data from “Job Rated Stress Score” from CareerCast.com). Do these data suggest that there is a correlation between job stress and annual income? Does it appear that jobs with more stress have higher salaries?

Stress	71.59	60.46	50.82	6.94	8.1	50.33	49.2	48.8	11.4
Median Salary	45.6	98.4	57	69	35.4	46.1	42.5	37.1	31.2

Short answer

There is not enough evidence to support the claim that there is a correlation between job stress and annual income.

As the two variables are not related, it cannot be concluded that jobs with more stress have higher salaries.

Step-by-step solution

Given information

Two samples are given showing the job stress score for the corresponding value of the median salary.

To test whether there is acorrelation between Stress and Median Salary or not.

Appropriate test

Since the correlation between the two variables need to be examined, the rank correlation test is the most appropriate non-parametric test.

Identify the hypothesis

The null hypothesis is as follows:

There is no correlation between the variables job stress and annual salary.

The alternative hypothesis is as follows:

There is a correlation between the variables job stress and annual salary.

Assign the ranks

Compute the ranks of each of the twosamples.

For a sample 1, assign rank 1 for the smallest observation, rank 2 to the next smallest observation, and so on until the largest observation.If some observations are equal, the mean of the ranks are assigned to each of the observations.

Similarly, assign ranks to the sample 2.

The following table shows the ranks of the 2 samples:

Ranks of Stress	9	8	7	1	2	6	5	4	3
Ranks of Median Salary	5	9	7	8	2	6	4	3	1

Spearman Rank Correlation Coefficient

Since there are no ties present, the following formula is used to compute the rank correlation coefficient:

\({r_s} = 1 - \frac{{6\sum {{d^2}} }}{{n\left( {{n^2} - 1} \right)}}\)

The table below shows the required calculations:

Ranks of Stress	9	8	7	1	2	6	5	4	3
Ranks of Median Salary	5	9	7	8	2	6	4	3	1
(difference) d	4	-1	0	-7	0	0	1	1	2
\({d^2}\)	16	1	0	49	0	0	1	1	4

The value of\(\sum {{d^2}} = 72\).

Here, n = 9.

Substituting the values in the formula, the value of\({r_s}\)is obtained as follows:

\(\begin{array}{c}{r_s} = 1 - \frac{{6\sum {{d^2}} }}{{n\left( {{n^2} - 1} \right)}}\\ = 1 - \frac{{6\left( {72} \right)}}{{9\left( {{9^2} - 1} \right)}}\\ = 0.4\end{array}\)

Therefore, the value of the spearman rank correlation coefficient is equal to 0.4.

Determine the critical value and the conclusion of the test

The critical values of the rank correlation coefficient for n=9 and\(\alpha = 0.05\)are -0.700 and 0.700.

If the value of the rank correlation coefficient falls between the critical values, the decision to reject the null hypothesis fails otherwise the null hypothesis is rejected.

Since the value of the rank correlation coefficientfalls in the interval bounded by the critical values, the null hypothesis is failed to reject.

There is not enough evidence to conclude that there is a correlation between job stress and annual income.

Since there is no correlation, it cannot be said that jobs with more stress have higher salaries.