Pizza and the Subway The “pizza connection” is the principle that the price of a slice of pizza in New York City is always about the same as the subway fare. Use the data listed below to determine whether there is a correlation between the cost of a slice of pizza and the subway fare.

Year	1960	1973	1986	1995	2002	2003	2009	2013	2015
Pizza Cost	0.15	0.35	1.00	1.25	1.75	2.00	2.25	2.3	2.75
Subway Fare	0.15	0.35	1.00	1.35	1.5	2.00	2.25	2.5	2.75
CPI	30.2	48.3	112.3	162.2	191.9	197.8	214.5	233	237.2

Data are given on the samples of the cost of pizza and subway fare.

The sample size (n) is 9.

The significance level is 0.05.

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

The null hypothesis is set up as follows:

There is no correlation between the cost of a pizza slice and the subway fare.

\({\rho _s} = 0\)

The alternative hypothesis is set up as follows:

There is a significant correlation between thecost of a pizza slice and the subway fare.

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

The test is two tailed.

Compute the ranks of each of the two samples as the data is not provided in the form of ranks.

Sort the data in ascending order for both samples. After that, assign ranks to both samples

In case some of the values are the same in the sample, the mean of the ranks is assigned to each of the observations.

The following table shows the ranks of the two samples:

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:

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.

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 fallin the interval bounded by the critical values, the null hypothesis is rejected.

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