Problem
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} What are the null and alternative hypotheses? A. $H_{0}: \rho=0$ $H_{1}: \rho \neq 0$ B. $H_{0}: \rho=0$ $H_{1}: \rho<0$ c. $H_{0}: \rho=0$ D. $H_{0}: \rho \neq 0$ $H_{1}: \rho>0$ $H_{1}: \rho=0$
Solution
Step 1 ：The null hypothesis is always a statement of no effect or no difference. In this case, it would be that there is no correlation between lemon imports and crash fatality rates.
Step 2 ：The alternative hypothesis is what we are testing for, which is that there is a correlation between lemon imports and crash fatality rates.
Step 3 ：Therefore, the correct hypotheses would be: \(H_{0}: \rho=0\) (There is no correlation between lemon imports and crash fatality rates) and \(H_{1}: \rho \neq 0\) (There is a correlation between lemon imports and crash fatality rates)
Step 4 ：So, the final answer is \(\boxed{A. H_{0}: \rho=0, H_{1}: \rho \neq 0}\)
