Question

CPI and the Subway Use CPI/subway data from the preceding exercise to test for a correlation between the CPI (Consumer Price Index) and the subway fare.

Short answer

The rank correlation coefficient between CPI and the subway fare is equal to 1.

There is enough evidence to conclude that there is a significant correlation between CPI and the subway fare.

Step-by-step solution

Given information

Data are provided on the samples of CPI and the subway fare.

The significance level is 0.05.

The paired set of values (n) is 9.

Identify the statistical hypothesis

Rank correlation coefficient is used to test the significance of the correlation between two ordinal variables.

The researcher wants to test for a correlation between the variables CPI and the subway free.

The null hypothesis is set up as follows:

There is no correlation between CPI and the subway fare.

\({\rho _s} = 0\)

The alternative hypothesis is set up as follows:

There is a significant correlation between CPI and the subway fare.

\({\rho _s} \ne 0\)

The test is two tailed.

Assign ranks

Compute the ranks of each of the twosamples.

Ranks are assigned from the smallest to the largest observations for the first sample. Similarly, assign ranks to the second sample.

If some observations are equal, the mean of the ranks isassigned to each of the observations.

The following table shows the ranks of the twosamples:

Ranks of Subway Fare	1	2	3	4	5	6	7	8	9
Ranks of CPI	1	2	3	4	5	6	7	8	9

Calculate the 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 following table shows the differences between the ranks of the two values for every pair:

Ranks of CPI	1	2	3	4	5	6	7	8	9
Ranks of Subway Fare	1	2	3	4	5	6	7	8	9
Difference (d)	0	0	0	0	0	0	0	0	0
\({d^2}\)	0	0	0	0	0	0	0	0	0

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

\(\begin{array}{c}{r_s} = 1 - \frac{{6\sum {{d^2}} }}{{n\left( {{n^2} - 1} \right)}}\\ = 1 - \frac{{6\left( 0 \right)}}{{9\left( {{9^2} - 1} \right)}}\\ = 1\end{array}\)

Therefore, the value of the Spearman rank correlation coefficient is equal to 1.

Obtain 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.

Since the value of the rank correlation coefficient does not fall in the interval bounded by the critical values, the null hypothesis is rejected.

There is enough evidence to conclude that there is a correlation between CPI and the subway fare.