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.)

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

Short answer

The scatterplot is shown below:

The value of the correlation coefficient is 0.113.

The p-value is 0.700.

There is not sufficient evidence to support the existence of a linear correlation between the heights of the president and the opponent.

No, they are not expected to be correlated as height is not one of the reasons for competing in elections.

Step-by-step solution

Given information

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

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

Sketch a scatterplot

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.

Compute the measure of the correlation coefficient

The correlation coefficient formula is

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

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

The valuesare tabulatedbelow:

x	y	\({x^2}\)	\({y^2}\)	\(xy\)
178	180	31684	32400	32040
182	180	33124	32400	32760
188	182	35344	33124	34216
175	173	30625	29929	30275
179	178	32041	31684	31862
183	182	33489	33124	33306
192	180	36864	32400	34560
182	180	33124	32400	32760
177	183	31329	33489	32391
185	177	34225	31329	32745
188	173	35344	29929	32524
188	188	35344	35344	35344
183	185	33489	34225	33855
188	175	35344	30625	32900
\(\sum x = 2568\)	\(\sum y = 2516\)	\(\sum {{x^2}} = 471370\)	\(\sum {{y^2} = } \;452402\)	\(\sum {xy\; = \;} 461538\)

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.

Step 4:Conduct a hypothesis test for correlation

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.

Discuss the expected correlation

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.