Question

Testing for Rank Correlation. In Exercises 7–12, use the rank correlation coefficient to test for a correlation between the two variables. Use a significance level of\(\alpha \)= 0.05.

Chocolate and Nobel Prizes The table below lists chocolate consumption (kg per capita) and the numbers of Nobel Laureates (per 10 million people) for several different countries (from Data Set 16 in Appendix B). Is there a correlation between chocolate consumption and the rate of Nobel Laureates? How could such a correlation be explained?

Chocolate	11.6	2.5	8.8	3.7	1.8	4.5	9.4	3.6	2	3.6	6.4
Nobel	12.7	1.9	12.7	3.3	1.5	11.4	25.5	3.1	1.9	1.7	31.9

Short answer

The rank correlation coefficient between the variables chocolate consumption and the number of Nobel Laureates is equal to 0.888.

There is enough evidence to conclude that there is a correlation between the variables chocolate consumption and the number of Laureates in the country.

The correlation between the given variables does not hold any meaning and thus can be called “spurious correlation”/”nonsense correlation.”

Step-by-step solution

Given information

Data are provided on the two samples for the variables chocolate consumption and the number of Laureates.

Determine the rank correlation test and the statistical hypothesis of the test

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

The null hypothesis is set up as follows:

There is no correlation between the variables chocolate consumption and the number of Laureates.

\({\rho _s} = 0\)

The alternative hypothesis is set up as follows:

There is a significant correlation between the variables chocolate consumption and the number of Laureates.

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

The test is twotailed.

Assign ranks

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

For sample 1, assign rank 1 for the smallest observation, rank 2 to the next smallest observation, and so on until the largest observation.Similarly, assign ranks for the second sample.

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

The following table shows the ranks of the two samples:

Ranks of Chocolate	11	3	9	6	1	7	10	4.5	2	4.5	8
Ranks of Laureates	8.5	3.5	8.5	6	1	7	10	5	3.5	2	11

Spearman rank correlation coefficient

Since there are ties present, the following formula is used to compute the rank correlation coefficient:

\({r_s} = \frac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}} \sqrt {n\left( {\sum {{y^2}} } \right) - {{\left( {\sum y } \right)}^2}} }}\)

Consider x to be the ranks assigned to sample 1 and y to be the ranks assigned to sample 2.

The table below shows the required calculations:

Ranks of chocolate(x)	Ranks of Laureates(y)	xy	\({x^2}\)	\({y^2}\)
11	8.5	93.5	121	72.25
3	3.5	10.5	9	12.25
9	8.5	76.5	81	72.25
6	6	36	36	36
1	1	1	1	1
7	7	49	49	49
10	10	100	100	100
4.5	5	22.5	20.25	25
2	3.5	7	4	12.25
4.5	2	9	20.25	4
8	11	88	64	121
\(\sum x \)=66	\(\sum y \)=66	\(\sum {xy} \)=493	\(\sum {{x^2}} \)=505.5	\(\sum {{y^2}} \)=505

Here, n = 11.

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

\(\begin{array}{c}{r_s} = \frac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}} \sqrt {n\left( {\sum {{y^2}} } \right) - {{\left( {\sum y } \right)}^2}} }}\\ = \frac{{11\left( {493} \right) - \left( {66} \right)\left( {66} \right)}}{{\sqrt {11\left( {505.5} \right) - {{\left( {66} \right)}^2}} \sqrt {11\left( {505} \right) - {{\left( {66} \right)}^2}} }}\\ = 0.888\end{array}\)

Therefore, the value of the Spearman rank correlation coefficient is equal to 0.888.

Determine the critical value and the conclusion of the test

The critical values of the rank correlation coefficient for n= 11 and\(\alpha = 0.05\)are -0.618 and 0.618.

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 the variables chocolate consumption and the number of Laureates in a country.

Meaning of correlation

Here, the value of the correlation indicates that as the per capita chocolate consumption increases, the number of Nobel Laureates also increases. This relation does not hold any meaning as the two variables are wildly apart and have no meaningful relationship in a real sense. Thus, this can be termed as meaningless or “spurious.”