Questions: Fifty-four wild bears were anesthetized, and then their weights and their 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 said, does it appear that a measured chest size can be used to predict the weight? Use a significance level of α=0.06. Correlation Results Correlation coeff, F: 0.963613 Critical r: ± 0.2680855 P-value (two tailed): 0.000 Determine the null and alternative hypotheses. H0 P H4 p

Fifty-four wild bears were anesthetized, and then their weights and their 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 said, does it appear that a measured chest size can be used to predict the weight? Use a significance level of α=0.06.

Correlation Results
Correlation coeff, F: 0.963613
Critical r: ± 0.2680855
P-value (two tailed): 0.000

Determine the null and alternative hypotheses.
H0 P
H4 p

Transcript text: Fifty-four wild bears were anesthetized, and then their weights and their 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 said, does it appear that a measured chest size can be used to predict the weight? Use a significance level of $\alpha=0.06$. \begin{tabular}{|l|l|} \hline \multicolumn{2}{|l|}{ Correlation Results } \\ \hline Correlation coeff, F: & 0.963613 \\ \hline Critical r: & $\pm 0.2680855$ \\ \hline P-value (two tailed): & 0.000 \\ \hline \end{tabular} Determine the null and alternative hypotheses. \[ \begin{array}{l} \mathrm{H}_{0} \mathrm{P} \square \square \\ \mathrm{H}_{4} \mathrm{p} \end{array} \]

Solution

Solution Steps

Step 1: Hypotheses Formulation

We define the null and alternative hypotheses as follows:

Null Hypothesis ($H_0$): There is no linear correlation between the weights of bears and their chest sizes.
Alternative Hypothesis ($H_1$): There is a linear correlation between the weights of bears and their chest sizes.

Step 2: Statistical Analysis

We have the following statistical values:

Correlation Coefficient ($r$): $0.9636$
Critical $r$: $\pm 0.2681$
P-value: $0.0000$
Significance Level ($\alpha$): $0.06$

Step 3: Significance Test

To determine if the correlation is statistically significant, we compare the absolute value of the correlation coefficient with the critical $r$: \[ |r| = 0.9636 > 0.2681 \] Since the absolute value of the correlation coefficient exceeds the critical $r$, we conclude that the correlation is statistically significant.

Step 4: Conclusion

Given that the P-value ($0.0000$) is less than the significance level ($\alpha = 0.06$), we reject the null hypothesis. Therefore, we have sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes.