Problem
Fifty-four wild bears were anesthetized, and then their weights and chest sizes were measured and listed in a data set. Results are shown in the accompanying display. Is there sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes? When measuring an anesthetized bear, is it easier to measure chest size than weight? If so, does it appear that a measured chest size can be used to predict the weight? Use a significance level of $\alpha=0.05$. Correlation Results \begin{tabular}{|l|l|} \hline Correlation coeff, $r$ : & 0.950441 \\ \hline Critical $r$ : & \pm 0.2680855 \\ \hline P-value (two tailed): & 0.000 \\ \hline \end{tabular} Determine the null and altemative hypotheses. \[ \begin{array}{l} H_{0}: \rho=0 \\ H_{1}: \rho \neq 0 \end{array} \] (Type integers or decimals. Do not round.) Identify the correlation coefficient, $r$. $r=\square$ (Round to three decimal places as needed.)
Solution
Step 1 ：Determine the null and alternative hypotheses. The null hypothesis \(H_{0}\) is that there is no correlation, i.e., \(\rho=0\). The alternative hypothesis \(H_{1}\) is that there is a correlation, i.e., \(\rho \neq 0\).
Step 2 ：Identify the correlation coefficient, \(r\). The given correlation coefficient is 0.950441.
Step 3 ：Round the correlation coefficient to three decimal places. The rounded correlation coefficient, \(r\), is 0.950.
Step 4 ：Final Answer: The correlation coefficient, \(r\), rounded to three decimal places is \(\boxed{0.950}\).
