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?
\begin{tabular}{|l|l|}
\hline \multicolumn{2}{|l|}{ Correlation Results } \\
\hline Correlation coeff, r: & 0.956999 \\
\hline Critical r: & \pm 0.2680855 \\
\hline P-value (two tailed): & 0.000 \\
\hline
\end{tabular}
If so, does it appear that a measured chest size can be used to predict the weight? Use a significance level of $\alpha=0.05$.
Determine the null and alternative hypotheses.
\[
\begin{array}{l}
\mathrm{H}_{0}: \rho=0 \\
\mathrm{H}_{1}: \rho \neq 0
\end{array}
\]
(Type integers or decimals. Do not round.)
Identify the correlation coefficient, $r$.
$r=.957$ (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 $r=\square$.
B. There are two critical values at $r= \pm \square$.
Answer
Final Answer: The correlation coefficient, \(r\), is \(\boxed{0.957}\). There are two critical values at \(r= \pm \boxed{0.268}\). There is sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes. Therefore, it appears that a measured chest size can be used to predict the weight of a bear. We reject the null hypothesis and accept the alternative hypothesis.
Step 1 ：Identify the null and alternative hypotheses. The null hypothesis, \(H_{0}\), is that there is no correlation (\(\rho=0\)), and the alternative hypothesis, \(H_{1}\), is that there is a correlation (\(\rho\neq0\)).
Step 2 ：Identify the correlation coefficient, \(r\). The correlation coefficient, \(r\), is given as 0.957, which is very close to 1. This indicates a strong positive linear correlation.
Step 3 ：Identify the critical value(s). The critical r values are given as ±0.2680855. The correlation coefficient is much larger than these critical values, indicating that the correlation is statistically significant.
Step 4 ：The p-value is given as 0.000, which is less than the significance level of 0.05, further supporting the claim of a significant correlation.
Step 5 ：Since the p-value is less than the significance level, we reject the null hypothesis and accept the alternative hypothesis.
Step 6 ：Therefore, it appears that a measured chest size can be used to predict the weight of a bear.
Step 7 ：Final Answer: The correlation coefficient, \(r\), is \(\boxed{0.957}\). There are two critical values at \(r= \pm \boxed{0.268}\). There is sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes. Therefore, it appears that a measured chest size can be used to predict the weight of a bear. We reject the null hypothesis and accept the alternative hypothesis.
