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.)

Manatees Listed below are numbers of registered pleasure boats in Florida (tens of thousands) and the numbers of manatee fatalities from encounters with boats in Florida for each of several recent years. The values are from Data Set 10 “Manatee Deaths” in Appendix B. Is there sufficient evidence to conclude that there is a linear correlation between numbers of registered pleasure boats and numbers of manatee boat fatalities?

Pleasure Boats	99	99	97	95	90	90	87	90	90
Manatee Fatalities	92	73	90	97	83	88	81	73	68

The data is listed for the number of registered pleasure boats and manatee fatalities in Florida.

A scatterplot determines the pattern in any paired set of data.

Steps to sketch a scatterplot:

1. Mark the two variables as pleasure boats and manatee fatalities on x and y-axis, respectively.
2. Mark the observations in a paired form on the graph.

The scatterplotis shown below.

The correlation coefficient formula is

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

Let variable x be pleasure boats counts and y be the count of manatee fatalities.

The valuesare listedin the table below:

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{9\left( {69407} \right) - \left( {837} \right)\left( {745} \right)}}{{\sqrt {9\left( {78005} \right) - {{\left( {837} \right)}^2}} \sqrt {9\left( {62449} \right) - {{\left( {745} \right)}^2}} }}\\ &= 0.341\end{aligned}\)

Thus, the correlation coefficient is 0.341.

Definemeasure\(\rho \)as the true correlation measure for the two variables.

For testing the claim, form the hypotheses:

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

The samplesize is9(n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.341}}{{\sqrt {\frac{{1 - {{\left( {0.341} \right)}^2}}}{{9 - 2}}} }}\\ &= 0.9597\\ &\approx 0.960\end{aligned}\)

Thus, the test statistic is 0.960.

The degree of freedom is calculated below:

\(\begin{aligned} df &= n - 2\\ &= 9 - 2\\ &= 7\end{aligned}\)

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

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T > 0.960} \right)\\ &= 0.369\end{aligned}\)

Thus, the p-value is 0.369.

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 a linear correlation between the counts of pleasure boats and manatee boat fatalities.