Question

Ages of Best Actresses and Best Actors Listed below are ages of Best Actresses and Best Actors at the times they won Oscars (from Data Set 14 “Oscar Winner Age” in Appendix B). Do these data suggest that there is a correlation between ages of Best Actresses and Best Actors?

Actress	61	32	33	45	29	62	22	44	54
Actor	45	50	48	60	50	39	55	44	33

Short answer

The rank correlation coefficient between the ages of actors and actresses is equal to -0.644

It can be concluded that there is no correlation between the ages of actors and actresses.

Step-by-step solution

Given information

Data are provided on the two samples on ages of actresses and actors.

Determine the rank correlation test and identify the statistical hypothesis

Rank correlation coefficient is a non-parametric test that is used to test the significance of the correlation between the two ordinal variables.

The null hypothesis is set up as follows:

There is no correlation between the ages of actresses and actors.

\({\rho _s} = 0\)

The alternative hypothesis is set up as follows:

There is a significant correlation between the ages of actresses and actors.

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

The test is twotailed.

Assign ranks

Compute the ranks of each of the two samples.

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

Similarly, assign ranks to the second sample.

The following table shows the ranks of the two samples:

Ranks of Ages of Actresses	8	3	4	6	2	9	1	5	7
Ranks of Ages of Actors	4	6.5	5	9	6.5	2	8	3	1

Calculate the Spearman rank correlation coefficient

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

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

The table below shows the required calculations:

Ranks of Ages of Actresses(x)	Ranks of Ages of Actors(y)	xy	\({x^2}\)	\({y^2}\)
8	4	32	64	16
3	6.5	19.5	9	42.25
4	5	20	16	25
6	9	54	36	81
2	6.5	13	4	42.25
9	2	18	81	4
1	8	8	1	64
5	3	15	25	9
7	1	7	49	1
\(\sum x \)=45	\(\sum y \)=45	\(\sum {xy} \)=186.5	\(\sum {{x^2}} \)=285	\(\sum {{y^2}} \)=284.5

Here, n = 9.

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

\(\begin{array}{c}{r_s} = \frac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}} \sqrt {n\left( {\sum {{y^2}} } \right) - {{\left( {\sum y } \right)}^2}} }}\\ = \frac{{9\left( {186.5} \right) - \left( {45} \right)\left( {45} \right)}}{{\sqrt {9\left( {285} \right) - {{\left( {45} \right)}^2}} \sqrt {9\left( {284.5} \right) - {{\left( {45} \right)}^2}} }}\\ = - 0.644\end{array}\)

Therefore, the value of the Spearman rank correlation coefficient is equal to -0.644.

Determine the critical value and 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 falls in the interval bounded by the critical values, so the decision if fail to reject the null hypothesis.

There is not enough evidence to conclude that there is a correlation between the ages of actresses and actors.