Question 3, 10.1.9
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Part 3 of 4
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Use the given data set to complete parts (a) through (c) below. (Use $\alpha=0.05$.)
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline $\mathbf{x}$ & 10 & 8 & 13 & 9 & 11 & 14 & 6 & 4 & 12 & 7 & 5 \\
\hline $\mathbf{y}$ & 9.14 & 8.14 & 8.74 & 8.78 & 9.26 & 8.09 & 6.14 & 3.09 & 9.14 & 7.26 & 4.74 \\
\hline
\end{tabular}
Click here to view a table of critical values for the correlation coefficient.
b. Find the linear correlation coefficient, r, then determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.
The linear correlation coefficient is $r=0.815$.
(Round to three decimal places as needed.)
Using the linear correlation coefficient found in the previous step, determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables. Choose the correct answer below.
A. There is sufficient evidence to support the claim of a nonlinear correlation between the two variables.
B. There is insufficient evidence to support the claim of a linear correlation between the two variables.
C. There is sufficient evidence to support the claim of a linear correlation between the two variables.
D. There is insufficient evidence to support the claim of a nonlinear correlation between the two variables.
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\(\boxed{\text{B. There is insufficient evidence to support the claim of a linear correlation between the two variables.}}\)
Step 1 ：The question is asking to determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables. The linear correlation coefficient, r, is given as 0.815. The value of r lies between -1 and 1. A value of 1 means a perfect positive correlation and a value of -1 means a perfect negative correlation. The closer the value of r to 1 or -1, the stronger the correlation. A value of 0 means no correlation. Here, r = 0.815 which is close to 1, indicating a strong positive correlation.
Step 2 ：However, to determine if there is sufficient evidence to support the claim of a linear correlation, we need to compare the calculated r value with the critical r value from the correlation coefficient table for a given significance level (alpha = 0.05) and degrees of freedom (n-2, where n is the number of pairs of data). If the calculated r value is greater than the critical r value, then there is sufficient evidence to support the claim of a linear correlation.
Step 3 ：The calculated r value (0.815) is less than the critical r value (2.262). Therefore, there is insufficient evidence to support the claim of a linear correlation between the two variables.
Step 4 ：\(\boxed{\text{B. There is insufficient evidence to support the claim of a linear correlation between the two variables.}}\)
