Listed below are annual data for various years. The data are weights (metric tons) of imported lemons and car crash fatality rates per 100,000 population. Construct a scatterplot, find the value of the linear correlation coefficient $r$, and find the P-value using $\alpha=0.05$. Is there sufficient evidence to conclude that there is a linear correlation between lemon imports and crash fatality rates? Do the results suggest that imported lemons cause car fatalities?
\begin{tabular}{|c|c|c|c|c|c|}
\hline Lemon Imports & 232 & 264 & 359 & 481 & 533 \\
\hline Crash Fatality Rate & 15.9 & 15.7 & 15.4 & 15.3 & 14.9 \\
\hline
\end{tabular}
The linear correlation coefficient is $r=$
(Round to three decimal places as needed.)
The test statistic is $\mathrm{t}=$
(Round to three decimal places as needed.)
The P-value is
(Round to three decimal places as needed.)
Answer
\(\boxed{\text{Final Answer: The linear correlation coefficient is } r = -0.959, \text{ the test statistic is } t = -5.872, \text{ and the P-value is } 0.010. \text{ There is a linear correlation between lemon imports and crash fatality rates.}}\)
Step 1 ：First, we need to calculate the sums and squares required for the Pearson's correlation coefficient formula. We also need to calculate the product of paired scores. The data for lemon imports are \([232, 264, 359, 481, 533]\) and for crash fatality rate are \([15.9, 15.7, 15.4, 15.3, 14.9]\). The number of pairs of scores is 5. The sum of lemon imports is 1869, the sum of crash fatality rate is 77.2, the sum of squares of lemon imports is 767851, the sum of squares of crash fatality rate is 1192.56, and the sum of the product of paired scores is 28663.2.
Step 2 ：Next, we substitute these values into the Pearson's correlation coefficient formula to get the correlation coefficient \(r\). The calculated \(r\) is -0.959.
Step 3 ：Then, we calculate the test statistic \(t\) using the formula given above. The calculated \(t\) is -5.872.
Step 4 ：Finally, we calculate the P-value. The calculated P-value is 0.010.
Step 5 ：Since the P-value is less than the significance level \(\alpha=0.05\), we reject the null hypothesis and conclude that there is a linear correlation between lemon imports and crash fatality rates.
Step 6 ：However, correlation does not imply causation, so we cannot conclude that imported lemons cause car fatalities.
Step 7 ：\(\boxed{\text{Final Answer: The linear correlation coefficient is } r = -0.959, \text{ the test statistic is } t = -5.872, \text{ and the P-value is } 0.010. \text{ There is a linear correlation between lemon imports and crash fatality rates.}}\)
