Provide an appropriate response.
A sports statistician is interested in determining if there is a relationship between the number of home team and visiting team losses and different sports. A random sample of 526 games is selected and the results are given below. Find the critical value $x_{\alpha}^{2}$ to test the claim that the number of home team and visiting team losses is independent of the sport. Use $a=0.01$.
\begin{tabular}{l|cccc}
& Football & Basketball & Soccer & Baseball \\
\hline Home team losses & 39 & 156 & 25 & 83 \\
Visiting team losses & 31 & 98 & 19 & 75
\end{tabular}
9.348
7.815
11.345
12.838
Answer
Final Answer: The critical value $x_{\alpha}^{2}$ to test the claim that the number of home team and visiting team losses is independent of the sport is \(\boxed{11.345}\).
Step 1 ：A sports statistician is interested in determining if there is a relationship between the number of home team and visiting team losses and different sports. A random sample of 526 games is selected and the results are given below. Find the critical value $x_{\alpha}^{2}$ to test the claim that the number of home team and visiting team losses is independent of the sport. Use $\alpha=0.01$.
Step 2 ：egin{tabular}{l|cccc} & Football & Basketball & Soccer & Baseball \\ \hline Home team losses & 39 & 156 & 25 & 83 \\ Visiting team losses & 31 & 98 & 19 & 75 \end{tabular}
Step 3 ：We can calculate the critical value for a given significance level and degrees of freedom using the chi-square distribution.
Step 4 ：Let's denote the significance level as $\alpha = 0.01$ and the degrees of freedom as $df = 3$.
Step 5 ：The critical value $x_{\alpha}^{2}$ can be calculated as $x_{\alpha}^{2} = \text{chi2.ppf}(1 - \alpha, df)$, where chi2.ppf is the percent point function (inverse of cdf — percentiles) of the chi-square distribution.
Step 6 ：Substituting the given values, we get $x_{\alpha}^{2} = \text{chi2.ppf}(1 - 0.01, 3)$.
Step 7 ：Calculating the above expression, we get $x_{\alpha}^{2} = 11.344866730144373$.
Step 8 ：Rounding to three decimal places, we get $x_{\alpha}^{2} = 11.345$.
Step 9 ：Final Answer: The critical value $x_{\alpha}^{2}$ to test the claim that the number of home team and visiting team losses is independent of the sport is \(\boxed{11.345}\).
