Question

using the following data set, run a linear regression to determine the linear regression coefficient. then, determine if theres a significant relationship between the variables.
critical values of the pearson
correlation coefficient r
r = 0.6241, there is a significant relationship between the variables.
r = 0.6241, there is not a significant relationship between the variables.
r = 0.5367, there is not a significant relationship between the variables.
r = 0.5367, there is a significant relationship between the variables.

Explanation:

Step1: Calculate the necessary sums

Let \(n = 10\) (number of data points).
Calculate \(\sum x=6 + 8+10 + 17+17+22+25+25+29+31=190\)
Calculate \(\sum y=47 + 46+90 + 56+61+74+53+60+224+164=875\)
Calculate \(\sum x^{2}=6^{2}+8^{2}+10^{2}+17^{2}+17^{2}+22^{2}+25^{2}+25^{2}+29^{2}+31^{2}=4080\)
Calculate \(\sum y^{2}=47^{2}+46^{2}+90^{2}+56^{2}+61^{2}+74^{2}+53^{2}+60^{2}+224^{2}+164^{2}=109341\)
Calculate \(\sum xy=(6\times47)+(8\times46)+(10\times90)+(17\times56)+(17\times61)+(22\times74)+(25\times53)+(25\times60)+(29\times224)+(31\times164)=29985\)

Step2: Calculate the Pearson correlation coefficient \(r\)

The formula for \(r\) is \(r=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}}\)
Substitute the values:
\(n = 10\), \(\sum x = 190\), \(\sum y=875\), \(\sum xy = 29985\), \(\sum x^{2}=4080\), \(\sum y^{2}=109341\)
First, calculate the numerator: \(n\sum xy-\sum x\sum y=10\times29985-190\times875=299850 - 166250=133600\)
Second, calculate \(n\sum x^{2}-(\sum x)^{2}=10\times4080-(190)^{2}=40800 - 36100 = 4700\)
Third, calculate \(n\sum y^{2}-(\sum y)^{2}=10\times109341-(875)^{2}=1093410-765625 = 327785\)
Then, \(\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}=\sqrt{4700\times327785}\approx\sqrt{1540589500}\approx39250\)
\(r=\frac{133600}{39250}\approx0.6241\)

Step3: Determine significance

For \(n = 10\), degrees of freedom \(df=n - 2=8\)
Looking at the critical - value table for \(\alpha = 0.05\), the critical value \(r_{c}=0.632\)
Since \(|r| = 0.6241<0.632\)

Answer:

\(r = 0.6241\), there is not a significant relationship between the variables.