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 $\alpha=0.05$.
\begin{tabular}{|l|l|l|}
\hline Correlation Results & \\
\hline Correlation coeff, $r:$ & 0.957422 \\
\hline Critical r: & \pm 0.2680855 \\
\hline P-value (two tailed): & 0.000 & \\
\hline
\end{tabular}
Determine the null and alternative 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, $\mathrm{r}$.
$r=0.957$ (Round to three decimal places as needed.)
Identify the critical value(s)
(Round to three decimal places as needed.)
A. There is one critical value at $r=$
B. There are two critical values at $r= \pm$.
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Answer
Since the correlation coefficient is much larger than the critical values, we can reject the null hypothesis and conclude that there is a significant correlation between the weights of bears and their chest sizes.
Step 1 ：The null and alternative hypotheses are: \[\begin{array}{l} H_{0}: \rho=0 \\ H_{1}: \rho \neq 0 \end{array}\]
Step 2 ：The correlation coefficient, r, is \(\boxed{0.957}\).
Step 3 ：There are two critical values at \(r= \pm \boxed{0.268}\).
Step 4 ：Since the correlation coefficient is much larger than the critical values, we can reject the null hypothesis and conclude that there is a significant correlation between the weights of bears and their chest sizes.
