Question

for the accompanying data set, (a) draw a scatter diagram of the data, (b) compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y. click here to view the data set. click here to view the critical values table. (a) draw a scatter diagram of the data. (b) compute the correlation coefficient. the correlation coefficient is r = 0.105. (round to three decimal places as needed.) (c) determine whether there is a linear relation between x and y. because the correlation coefficient is and the absolute value of the correlation coefficient, is than the critical value for linear relation exists between x and y. (round to three decimal places as ne

Explanation:

Step1: Recall the formula for the correlation coefficient

The formula for the correlation coefficient \(r=\frac{n\sum xy-\sum x\sum y}{\sqrt{n\sum x^{2}-(\sum x)^{2}}\sqrt{n\sum y^{2}-(\sum y)^{2}}}\)

First, calculate \(\sum x\), \(\sum y\), \(\sum xy\), \(\sum x^{2}\), \(\sum y^{2}\)
Given \(x = [7,6,1,7,9]\), \(\sum x=7 + 6+1+7+9=30\)
Given \(y = [8,7,6,9,5]\), \(\sum y=8 + 7+6+9+5=35\)
\(\sum xy=(7\times8)+(6\times7)+(1\times6)+(7\times9)+(9\times5)=56 + 42+6+63+45 = 212\)
\(\sum x^{2}=7^{2}+6^{2}+1^{2}+7^{2}+9^{2}=49+36 + 1+49+81=216\)
\(\sum y^{2}=8^{2}+7^{2}+6^{2}+9^{2}+5^{2}=64+49+36+81+25 = 255\)

Step2: Substitute into the formula

\(n = 5\)
\(r=\frac{5\times212-30\times35}{\sqrt{5\times216-30^{2}}\sqrt{5\times255-35^{2}}}\)
\(=\frac{1060 - 1050}{\sqrt{1080-900}\sqrt{1275 - 1225}}\)
\(=\frac{10}{\sqrt{180}\sqrt{50}}\)
\(=\frac{10}{\sqrt{9000}}\)
\(=\frac{10}{94.868}\approx0.105\)

Step3: Check for linear relation

The critical value for \(n = 5\) (degrees of freedom \(n - 2=3\)) at a common significance level (e.g., \(\alpha = 0.05\)) is \(r_{c}=0.878\)
Since \(|r|=|0.105|<0.878\)

Answer:

Because the correlation coefficient is positive and the absolute value of the correlation coefficient, \(0.105\), is less than the critical value for linear relation exists between \(x\) and \(y\) (i.e., there is no linear relation).