Question

use the given data set to complete parts (a) through (c) below. (use α = 0.05.)
x: 10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5
y: 9.14, 8.14, 8.74, 8.76, 9.26, 8.11, 6.14, 3.09, 9.13, 7.27, 4.74
click here to view a table of critical values for the correlation coefficient.
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 insufficient evidence to support the claim of a nonlinear correlation between the two variables.
b. there is sufficient evidence to support the claim of a linear correlation between the two variables.
c. there is insufficient evidence to support the claim of a linear correlation between the two variables.
d. there is sufficient evidence to support the claim of a nonlinear correlation between the two variables.

Explanation:

To determine the correct answer, we analyze the options:

• Option A and D are about nonlinear correlation, but the question is about linear correlation, so they are incorrect.
• For linear correlation, we compare the calculated correlation coefficient with the critical value (from the table, with \(n = 11\) and \(\alpha=0.05\), critical value \(r_{critical}\approx0.602\)). First, calculate the correlation coefficient \(r\) for the data. Using the formula \(r=\frac{n\sum xy - \sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\):
• \(\sum x = 10 + 8+13 + 9+11 + 14+6 + 4+12 + 7+5=99\)
• \(\sum y=9.14 + 8.14+8.74 + 8.76+9.26 + 8.11+6.14 + 3.09+9.13 + 7.27+4.74 = 78.52\)
• \(\sum xy=10\times9.14 + 8\times8.14+\cdots+5\times4.74 = 703.46\)
• \(\sum x^{2}=10^{2}+8^{2}+\cdots+5^{2}=969\)
• \(\sum y^{2}=9.14^{2}+8.14^{2}+\cdots+4.74^{2}=599.34\)
• \(n = 11\)
• Plugging in: \(r=\frac{11\times703.46-99\times78.52}{\sqrt{[11\times969 - 99^{2}][11\times599.34 - 78.52^{2}]}}\approx0.81\)
• Since \(|r| = 0.81>0.602\) (critical value), we reject the null hypothesis (no linear correlation) and conclude there is sufficient evidence for linear correlation. So option B is correct.

Answer:

B. There is sufficient evidence to support the claim of a linear correlation between the two variables.