91影视

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鈥檚 height and y as the opponent鈥檚 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鈥檚 and the opponent鈥檚 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.