Question

Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Oscars Listed below are ages of Oscar winners matched by the years in which the awards were won (from Data Set 14 “Oscar Winner Age” in Appendix B). Is there sufficient evidence to conclude that there is a linear correlation between the ages of Best Actresses and Best Actors? Should we expect that there would be a correlation?

Actress	28	30	29	61	32	33	45	29	62	22	44	54
Actor	43	37	38	45	50	48	60	50	39	55	44	33

Short answer

The scatter plot is shown below:

The value of the correlation coefficient is –0.288.

The p-value is 0.364.

There is not enough evidence to support the claim that there existsa linear correlation between the ages of actresses and actors.

No correlation is expected between the ages of best actressesand actors in Oscars.

Step-by-step solution

Given information

The data is recorded for the ages of best actresses and actorsin Oscars.

Best Actress	Best Actor
28	43
30	37
29	38
61	45
32	50
33	48
45	60
29	50
62	39
22	55
44	44
54	33

Sketch a scatterplot

A scatterplot is used to reveal the pattern of association between two variables.

Steps to sketch a scatterplot:

1. Define two axesfor the ages of actors and actresseseach year.
2. Mark each paired set of values on the graph.

The resultant scatterplot is shown below.

Compute the measure of the correlation coefficient

The correlation coefficient formula is

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

For the variable of age for best actress (x) and age of best actor (y), compute the correlation coefficient.

The valuesare listedin the table below:

x	y	\({x^2}\)	\({y^2}\)	\(xy\)
28	43	784	1849	1204
30	37	900	1369	1110
29	38	841	1444	1102
61	45	3721	2025	2745
32	50	1024	2500	1600
33	48	1089	2304	1584
45	60	2025	3600	2700
29	50	841	2500	1450
62	39	3844	1521	2418
22	55	484	3025	1210
44	44	1936	1936	1936
54	33	2916	1089	1782
\(\sum x = 469\)	\(\sum y = 542\)	\(\sum {{x^2}} = 20405\)	\(\sum {{y^2} = } \;25162\)	\(\sum {xy\; = \;} 20841\)

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{12\left( {20841} \right) - \left( {469} \right)\left( {542} \right)}}{{\sqrt {12\left( {20405} \right) - {{\left( {469} \right)}^2}} \sqrt {12\left( {25162} \right) - {{\left( {542} \right)}^2}} }}\\ &= - 0.288\end{aligned}\)

Thus, the correlation coefficient is –0.288.

Step 4:Conduct a hypothesis test for correlation

Definethe actual measure of the correlation coefficient between the age of best actress and actor in Oscars as\(\rho \).

For testing the claim, form the hypotheses:

\(\begin{array}{l}{H_o}:\rho = 0\\{H_a}:\rho \ne 0\end{array}\)

The samplesize is 12 (n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{ - 0.288}}{{\sqrt {\frac{{1 - {{\left( { - 0.288} \right)}^2}}}{{12 - 2}}} }}\\ &= - 0.951\end{aligned}\)

Thus, the test statistic is–0.951.

The degree of freedom is

\(\begin{aligned} df &= n - 2\\ &= 12 - 2\\ &= 10.\end{aligned}\)

The p-value is computed from the t-distribution table.

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T < - 0.951} \right)\\ &= 0.364\end{aligned}\)

Thus, the p-value is 0.364.

Since the p-value is greater than 0.05, the null hypothes is fails to berejected.

Therefore, there is not enough evidence to conclude that there is no linear correlation between the age of actors and actresses.

Discuss if there exists a correlation between the ages of actors and actresses

It was not expected that actors and actresses have acorrelation between their ages.They can beparts of different movies.There is no clear logic for any association between their ages.