Question

analyzing regression results
latasha was presented with the following data set and argued that there was no correlation between x and y. is latasha correct? use the regression equation to explain your reasoning.

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

The mean of \(x\) values: \(\bar{x}=\frac{1 + 2+3+4+5+6+7}{7}=\frac{28}{7} = 4\)
The mean of \(y\) values: \(\bar{y}=\frac{4 + 5+4+5+4+5+4}{7}=\frac{31}{7}\approx4.43\)

Step2: Calculate the slope \(b\) of the regression line

The formula for \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

\(\sum_{i = 1}^{7}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 4)(4 - 4.43)+(2 - 4)(5 - 4.43)+(3 - 4)(4 - 4.43)+(4 - 4)(5 - 4.43)+(5 - 4)(4 - 4.43)+(6 - 4)(5 - 4.43)+(7 - 4)(4 - 4.43)\)
\(=(- 3)(-0.43)+(-2)(0.57)+(-1)(-0.43)+(0)(0.57)+(1)(-0.43)+(2)(0.57)+(3)(-0.43)\)
\(=1.29-1.14 + 0.43+0-0.43+1.14-1.29=0\)

\(\sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}=(1 - 4)^{2}+(2 - 4)^{2}+(3 - 4)^{2}+(4 - 4)^{2}+(5 - 4)^{2}+(6 - 4)^{2}+(7 - 4)^{2}\)
\(=9 + 4+1+0+1+4+9=28\)

Since \(b = 0\), the regression equation is \(y=\bar{y}\) (because \(y=b(x-\bar{x})+\bar{y}\), and with \(b = 0\), \(y=\bar{y}\))

Answer:

LaTasha is correct. Since the slope \(b\) of the regression line (calculated using the formula \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)) is \(0\), there is no linear correlation between \(x\) and \(y\). A slope of \(0\) for the regression line \(y = mx + c\) (in the case of simple linear regression) implies that changes in \(x\) do not lead to a linear - change in \(y\) (i.e., no linear relationship).