Let $x$ be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let $y$ be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of $n=6$ professional basketball players gave the following information. \begin{tabular}{l|llllll} \hline$x$ & 64 & 64 & 88 & 66 & 64 & 72 \\ \hline$y$ & 38 & 45 & 50 & 45 & 45 & 41 \\ \hline \end{tabular} Given that $\sum x=418, \sum y=264, \sum x^{2}=29,572, \sum y^{2}=11,700, \sum x y=18,514$, and $r=0.627$, find the critical value for a test using a $2.5 \%$ level of significance claiming that $\varrho$ is greater than zero. 2.776 0.741 4.604 0.727 6.869
Solution
Step 1 ：Let $x$ be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let $y$ be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of $n=6$ professional basketball players gave the following information.
Step 2 ：\begin{tabular}{l|llllll}\n\hline$x$ & 64 & 64 & 88 & 66 & 64 & 72 \\n\hline$y$ & 38 & 45 & 50 & 45 & 45 & 41 \\n\hline\n\end{tabular}
Step 3 ：Given that $\sum x=418, \sum y=264, \sum x^{2}=29,572, \sum y^{2}=11,700, \sum x y=18,514$, and $r=0.627$, we are asked to find the critical value for a test using a $2.5 \%$ level of significance claiming that $\varrho$ is greater than zero.
Step 4 ：The number of pairs of data is $n = 6$.
Step 5 ：The degrees of freedom is calculated as $df = n - 2 = 6 - 2 = 4$.
Step 6 ：The level of significance is given as $\alpha = 0.025$.
Step 7 ：Since this is a one-tailed test, we use the percent point function (ppf) to find the critical value.
Step 8 ：The critical value is calculated as $2.7764451051977987$.
Step 9 ：Rounding to three decimal places, the critical value is $2.776$.
Step 10 ：Final Answer: The critical value for a test using a 2.5% level of significance claiming that the population correlation coefficient (ρ) is greater than zero is \(\boxed{2.776}\).
