The weights (in pounds) of elght vehicles and the variabilities of their braking distances (in feet) when stopping on a wet surface are shown in the table. At $\alpha=0.01$, is there enough evidence to conclude that there is a significant linear cortclation batween vehicle weight and variability in braking distance on a wet surface? \begin{tabular}{|l|c|c|c|c|c|c|c|c|} \hline Weight, $\mathrm{x}$ & 5970 & 5360 & 6500 & 5100 & 5870 & 4800 & 5700 & 5880 \\ \hline Variability, $\mathrm{y}$ & 293 & 240 & 4.07 & 178 & 294 & 249 & 2.25 & 270 \\ \hline \end{tabular} Setup the hypothesis for the test \[ \begin{array}{l} H_{0} p=0 \\ H_{a} p=0 \end{array} \] Calculate the test statistic $t=33$ (Round to two decimal places as needed) Cilculate the Pivalue \[ \text { P.value }=\square \text { (Round to three decirtial places as needect) } \]
Solution
Step 1 ：Setup the hypothesis for the test: \(H_{0}: p=0\) (There is no correlation) and \(H_{a}: p \neq 0\) (There is a correlation).
Step 2 ：Calculate the test statistic: \(t=33\).
Step 3 ：Calculate the degrees of freedom for this test: \(df = n-2 = 8-2 = 6\).
Step 4 ：Calculate the p-value using the t-distribution with the given test statistic and degrees of freedom: \(p = 5.151762394461912 \times 10^{-8}\).
Step 5 ：Compare the p-value with the significance level \(\alpha = 0.01\). Since the p-value is much less than the significance level, we reject the null hypothesis.
Step 6 ：Conclude that there is a significant linear correlation between vehicle weight and variability in braking distance on a wet surface.
Step 7 ：Final Answer: The p-value is \(\boxed{5.15 \times 10^{-8}}\).
