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.)

POTUS Media periodically discuss the issue of heights of winning presidential candidates and heights of their main opponents. Listed below are those heights (cm) from severalrecent presidential elections (from Data Set 15 “Presidents” in Appendix B). Is there sufficient evidence to conclude that there is a linear correlation between heights of winning presidential candidates and heights of their main opponents? Should there be such a correlation?

President	178	182	188	175	179	183	192	182	177	185	188	188	183	188
Opponent	180	180	182	173	178	182	180	180	183	177	173	188	185	175

The data is listedfor the heights of presidential candidates and their opponents.

A scatterplot describes a trend for two variables recorded in the paired form.

Steps to sketch a scatterplot:

1. Describe theaxes for the height of presidents and opponents.
2. Mark dots for paired observations.

The resultantscatterplotis shown below.

The correlation coefficient formula is

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

Define x as the president’s height and y as the opponent’s height.

The valuesare tabulatedbelow:

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{14\left( {461538} \right) - \left( {2568} \right)\left( {2516} \right)}}{{\sqrt {14\left( {471370} \right) - {{\left( {2568} \right)}^2}} \sqrt {14\left( {452402} \right) - {{\left( {2516} \right)}^2}} }}\\ &= 0.113\end{aligned}\)

Thus, the correlation coefficient is 0.113.

Definethe measure\(\rho \)as the linear correlation between two variables:the height of the president and the opponent.

For testing the claim, form the hypotheses:

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

The samplesize is14(n).

The test statistic is calculated below:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.113}}{{\sqrt {\frac{{1 - {{\left( {0.113} \right)}^2}}}{{14 - 2}}} }}\\ &= 0.394\end{aligned}\)

Thus, the test statistic is 0.394.

The degree of freedom iscalculated below:

\(\begin{aligned} df &= n - 2\\ &= 14 - 2\\ &= 12\end{aligned}\)

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

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T > 0.394} \right)\\ &= 0.700\end{aligned}\)

Thus, the p-value is 0.700.

Since thep-value is greater than 0.05, the null hypothesis fails to be rejected.

Therefore, there is not sufficient evidence to conclude the existence of alinear correlation between the president’s and the opponent’s height.

The heights of presidents and opponents are expected to be uncorrelated. One possible reason is that the candidates who participate inelections are expected to compete for more important reasons than their height.