Question

use the given data set to complete parts (a) through (c) below. (use α = 0.05)

x	y
8	6.76
13	12.74
9	7.11
11	7.82
14	8.84
6	6.08
4	5.38
12	8.16
7	6.43
5	5.73

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 = 0.817 (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 insufficient evidence to support the claim of a linear 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 sufficient evidence to support the claim of a nonlinear correlation between the two variables.
d. there is insufficient evidence to support the claim of a nonlinear correlation between the two variables.

Explanation:

Step1: Recall correlation coefficient interpretation

The linear correlation coefficient \( r = 0.817 \) (close to 1, indicating a strong positive linear relationship). For \( \alpha = 0.05 \), we compare \( |r| \) with the critical value. If \( |r| \) exceeds the critical value, there's sufficient evidence for linear correlation.

Step2: Analyze the options

• Option A: Says insufficient for linear, but \( r = 0.817 \) is strong, so A is wrong.
• Option B: States sufficient evidence for linear correlation. Since \( r \) is significant (close to 1, and we assume critical value for \( n \) (number of data points: let's count \( x \) values: 10,8,13,9,11,14,6,4,12,7,5 – 11 points? Wait, critical value for \( n = 11 \) (wait, count the \( x \) entries: 10,8,13,9,11,14,6,4,12,7,5 – 11 data points. Critical value for \( n = 11 \) and \( \alpha = 0.05 \) is around 0.602 (from critical value table). \( |0.817| > 0.602 \), so sufficient evidence for linear correlation. So B is correct.
• Option C: Talks about nonlinear, but we're testing linear correlation, so C is wrong.
• Option D: Says insufficient, but we have sufficient, so D is wrong.

Answer:

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