Question

Interpreting a Computer Display. In Exercises 9–12, refer to the display obtained by using the paired data consisting of Florida registered boats (tens of thousands) and numbers of manatee deaths from encounters with boats in Florida for different recent years (from Data Set 10 in Appendix B). Along with the paired boat, manatee sample data, Stat Crunch was also given the value of 85 (tens of thousands) boats to be used for predicting manatee fatalities.

Testing for Correlation Use the information provided in the display to determine the value of the linear correlation coefficient. Is there sufficient evidence to support a claim of a linear correlation between numbers of registered boats and numbers of manatee deaths from encounters with boats?

Short answer

The linear correlation coefficient is 0.85014394.

There is sufficient evidence to support the claim that the variables “number of registered boats” and “number of manatee deaths from encounters with boats” are linearly related.

Step-by-step solution

Given information

Results are obtained for the linear relation between the variables “number of registered boats” and “number of manatee deaths” using StatCrunch.

Correlation coefficient

The linear correlation between the two variables is shown in the results obtained using StatCrunch and is equal to 0.85014394.

Significance of correlation

The researcher wants to test the claim that there is a linear correlation between the variables number of registered boats and the number of manatee deaths from encounters with boats.

Let\(\rho \)represent the linear correlation coefficient of the population.

The statistical hypothesis is given below:

\(\begin{array}{l}{H_0}:\rho = 0\\{H_1}:\rho \ne 0\end{array}\)

Here,

\(\begin{array}{l}r = 0.850\\n = 24\end{array}\)

The degrees of freedom (df) is given as follows:

\(\begin{array}{c}df = n - 2\\ = 24 - 2\\ = 22\end{array}\)

The two-tailed critical values of rat df = 22 and at a significance level of 0.05 are obtained as –404 and 0.404.

Since the given value of the linear correlation coefficient (0.850) lies beyond the upper critical value of 0.404, the null hypothesis is rejected.

Thus, there issufficient evidence to support the claim of a linear correlation between the numbers of registered boats and number of manatee deaths from encounters with boats.