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	6.09
4	5.39
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. (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.
c. identify the feature of the data that would be missed if part (b) was completed without constructing the scatterplot. choose the correct answer below.
a. the scatterplot does not reveal a perfect straight-line pattern.
b. the scatterplot reveals a perfect straight-line pattern and does not contain any outliers.
c. the scatterplot reveals a perfect straight-line pattern, except for the presence of one outlier.
d. the scatterplot does not reveal a perfect straight-line pattern, and contains one outlier.

Explanation:

Step1: Recall Linear Correlation Test

To determine if there's sufficient evidence for a linear correlation, we compare the absolute value of the correlation coefficient \(|r|\) with the critical value (from the table, for \(n\) data points and \(\alpha = 0.05\)). First, find \(n\) (number of data points). From the table, \(x\) values: 10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5. So \(n = 11\).

Step2: Find Critical Value

For \(n = 11\) and \(\alpha = 0.05\), the critical value for the correlation coefficient (from critical value table) is approximately \(0.602\) (standard critical values for Pearson's \(r\): for \(n = 11\), critical value \(r_{critical} \approx 0.602\)).

Step3: Compare \(|r|\) and Critical Value

Given \(r = 0.817\), \(|r| = 0.817\). Since \(0.817 > 0.602\) (critical value), we reject the null hypothesis (no linear correlation) and conclude there is sufficient evidence for a linear correlation.

Step4: Analyze Scatterplot Feature (Part c)

Looking at the data, when we plot \(x\) and \(y\), most points follow a straight-line pattern, but there's one outlier (checking the data: e.g., \(x = 13\), \(y = 12.74\) or others? Wait, the option C says "The scatterplot reveals a perfect straight-line pattern, except for the presence of one outlier". Wait, no—wait, the linear correlation is strong (\(r = 0.817\)), but not perfect. Wait, the options: Option C: "The scatterplot reveals a perfect straight-line pattern, except for the presence of one outlier"—no, wait, maybe I misread. Wait, the correct analysis: when \(r\) is close to 1 but not 1, the scatterplot has a strong linear pattern with maybe one outlier. Wait, the data points: let's list \(x\) and \(y\):

\(x\): 10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5

\(y\): 7.46, 6.76, 12.74, 7.11, 7.82, 8.84, 6.09, 5.39, 8.16, 6.43, 5.73

Plotting these, most points show a linear trend, but one point (maybe \(x = 13\), \(y = 12.74\) or \(x = 4\), \(y = 5.39\)?) is an outlier. The option C says "The scatterplot reveals a perfect straight-line pattern, except for the presence of one outlier"—wait, no, \(r = 0.817\) is not perfect (1), but the option C: Wait, maybe the correct option for part c is C? Wait, no, let's recheck.

Wait, part b: the correct answer for the correlation test is B: "There is sufficient evidence to support the claim of a linear correlation between the two variables" because \(|r| = 0.817 > r_{critical} \approx 0.602\) (for \(n = 11\), \(\alpha = 0.05\)).

For part c: the scatterplot has a strong linear pattern (since \(r\) is high) but with one outlier. So option C: "The scatterplot reveals a perfect straight-line pattern, except for the presence of one outlier"—wait, no, "perfect" is not right, but maybe the option is C. Wait, the options:

A. The scatterplot does not reveal a perfect straight-line pattern.

B. The scatterplot reveals a perfect straight-line pattern and does not contain any outliers.

C. The scatterplot reveals a perfect straight-line pattern, except for the presence of one outlier.

D. The scatterplot does not reveal a perfect straight-line pattern, and contains one outlier.

Wait, \(r = 0.817\) is close to 1, so the pattern is nearly perfect but with one outlier. So option C.

Answer:

(Part b): B. There is sufficient evidence to support the claim of a linear correlation between the two variables