A case-control (or retrospective) study was conducted to investigate a relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in a crash. Results are given in the accompanying table. Using a 0.01 significance level, test the claim that injuries are independent of helmet color. \begin{tabular}{l|ccccc} & \multicolumn{5}{|c}{ Color of Helmet } \\ \cline { 2 - 6 } & Black & White & Yellow & Red & Blue \\ \hline Controls (not injured) & 500 & 352 & 34 & 164 & 86 \\ \begin{tabular}{l} Cases (injured \\ or killed) \end{tabular} & 222 & 114 & 9 & 68 & 42 \end{tabular} Click here to view the chi-square distribution table. A. $\mathrm{H}_{0}$ : Injuries and helmet color are independent $\mathrm{H}_{1}$ : Injuries and helmet color are dependent B. $\mathrm{H}_{0}$ : Whether a crash occurs and helmet color are independent $\mathrm{H}_{1}$ : Whether a crash occurs and helmet color are dependent c. $\mathrm{H}_{0}$ : Whether a crash occurs and helmet color are dependent $\mathrm{H}_{1}$ : Whether a crash occurs and helmet color are independent D. $\mathrm{H}_{0}$ : Injuries and helmet color are dependent $\mathrm{H}_{1}$ : Injuries and helmet color are independent Compute the test statistic. (Round to three decimal places as needed.)

Step 1 ：State the hypotheses. The null hypothesis (H0) is that injuries and helmet color are independent, and the alternative hypothesis (H1) is that injuries and helmet color are dependent. \[H0: \text{Injuries and helmet color are independent}\] \[H1: \text{Injuries and helmet color are dependent}\]

Step 2 ：Calculate the expected frequencies for each cell. The expected frequency is calculated as \((\text{row total} \times \text{column total}) / \text{grand total}\). The expected frequencies are calculated as follows: \[\text{Black (Controls)}: (1136\times722)/1591 = 513.02\] \[\text{Black (Cases)}: (455\times722)/1591 = 208.98\] \[\text{White (Controls)}: (1136\times466)/1591 = 332.91\] \[\text{White (Cases)}: (455\times466)/1591 = 133.09\] \[\text{Yellow (Controls)}: (1136\times43)/1591 = 30.77\] \[\text{Yellow (Cases)}: (455\times43)/1591 = 12.23\] \[\text{Red (Controls)}: (1136\times232)/1591 = 166.32\] \[\text{Red (Cases)}: (455\times232)/1591 = 65.68\] \[\text{Blue (Controls)}: (1136\times128)/1591 = 91.38\] \[\text{Blue (Cases)}: (455\times128)/1591 = 36.62\]

Step 3 ：Compute the test statistic. The test statistic is the sum of \((\text{observed}-\text{expected})^2 / \text{expected}\) for each cell. The test statistic is calculated as follows: \[\chi^2 = \sum \frac{(O-E)^2}{E} = 7.36\]

Step 4 ：Determine the critical value. With a 0.01 significance level and degrees of freedom (df) = (number of rows - 1) * (number of columns - 1) = (2-1)*(5-1) = 4, the critical value from the chi-square distribution table is 13.28.

Step 5 ：Make a decision. Since the test statistic (7.36) is less than the critical value (13.28), we do not reject the null hypothesis. Therefore, we conclude that injuries and helmet color are independent at the 0.01 significance level. \[\boxed{\text{Injuries and helmet color are independent at the 0.01 significance level.}}\]