91影视

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.