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 9.15 8.15 8.74 8.78 9.27 8.09 6.12 3.09 9.13 7.27 4.74
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= 0.814.
(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 linear 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 insufficient evidence to support the claim of a nonlinear correlation between the two variables.
d. there is sufficient evidence to support the claim of a nonlinear correlation between the two variables.

Explanation:

Step1: Determine sample size

The data set has \( n = 11 \) pairs of \( x \) and \( y \) values.

Step2: Find critical value

Using the table of critical values for the correlation coefficient with \( \alpha = 0.05 \) and \( n = 11 \), the critical value \( r_{critical} \) is approximately \( 0.602 \) (from standard critical value tables for correlation).

Step3: Compare \( |r| \) and \( r_{critical} \)

We have \( r = 0.814 \), so \( |r| = 0.814 \). Since \( 0.814> 0.602 \) (i.e., \( |r|>r_{critical} \)), we reject the null hypothesis (of no linear correlation) and conclude there is sufficient evidence for a linear correlation. Also, we are testing for linear correlation, so options C and D (about nonlinear) are incorrect. Since \( |r| \) exceeds the critical value, there is sufficient evidence.

Answer:

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