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.

Student and U.S. News & World Report Rankings of Colleges Each year, U.S. News & World Report publishes rankings of colleges based on statistics such as admission rates, graduation rates, class size, faculty-student ratio, faculty salaries, and peer ratings of administrators. Economists Christopher Avery, Mark Glickman, Caroline Minter Hoxby, and Andrew Metrick took an alternative approach of analyzing the college choices of 3240 high-achieving school seniors. They examined the colleges that offered admission along with the colleges that the students chose to attend. The table below lists rankings for a small sample of colleges. Find the value of the rank correlation coefficient and use it to determine whether there is a correlation between the student rankings and the rankings of the magazine.

Student ranks	1	2	3	4	5	6	7	8
U.S. News&World Report ranks	1	2	5	4	7	6	3	8

Short answer

The rank correlation coefficient is 0.714.

It can be concluded that there is not a significant correlation between the student rankings and the rankings of the magazine.

Step-by-step solution

Given information

Three samples show the rankings of colleges by students and by a magazine.

Identify the hypothesis of the test

The rank correlation test is used to test the significance of the rank correlation between the given two variables.

The null hypothesis is as follows:

There is no correlation between the student rankings and the rankings of the magazine, i.e.,\({\rho _s} = 0\).

The alternative hypothesis is as follows:

There is a correlation between the student rankings and the rankings of the magazine,i.e.,\({\rho _s} \ne 0\).

Differences betweenthe ranks

Compute the differences betweenthe rankings by subtracting the rankings by the magazine from those by the students.

Compute the square of the differences.

The following table shows the differences as well as their squared values:

Student ranks	1	2	3	4	5	6	7	8
U.S. Newsand World Report ranks	1	2	5	4	7	6	3	8
Differences (d)	0	0	-2	0	-2	0	4	0
\({d^2}\)	0	0	4	0	4	0	16	0

The sum of the squared differences is computed below:

\(\begin{array}{c}\sum {{d^2}} = 0 + 0 + 4 + 0 + 4 + 0 + 16 + 0\\ = 24\end{array}\)

Calculate the rank correlation coefficient

Here, n is equal to 8.

Compute the rank correlation coefficient using the given formula:

\(\begin{array}{c}{r_s} = 1 - \frac{{6\sum {{d^2}} }}{{n\left( {{n^2} - 1} \right)}}\\ = 1 - \frac{{6\left( {24} \right)}}{{8\left( {{8^2} - 1} \right)}}\\ = 0.71429\end{array}\)

Thus, the rank correlation coefficient is 0.71429.

Determine the critical value and the conclusion of the test

Since nis less than 30, the critical values of\({r_s}\)at\(\alpha = 0.05\)andn= 8are–0.738 and 0.738.

As the value of\({r_s}\)lies between the critical values, the null hypothesis fails to reject.

There is not enough evidence to conclude that there is a correlation between the rankings of the students and of the magazine.