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 \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(\sum_{i = 1}^{7}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 4)(4-\frac{31}{7})+(2 - 4)(5-\frac{31}{7})+(3 - 4)(4-\frac{31}{7})+(4 - 4)(5-\frac{31}{7})+(5 - 4)(4-\frac{31}{7})+(6 - 4)(5-\frac{31}{7})+(7 - 4)(4-\frac{31}{7})\)
\(=(- 3)(\frac{28 - 31}{7})+(-2)(\frac{35 - 31}{7})+(-1)(\frac{28 - 31}{7})+0\times(\frac{35 - 31}{7})+1\times(\frac{28 - 31}{7})+2\times(\frac{35 - 31}{7})+3\times(\frac{28 - 31}{7})\)
\(=(-3)\times(-\frac{3}{7})+(-2)\times\frac{4}{7}+(-1)\times(-\frac{3}{7})+0 + 1\times(-\frac{3}{7})+2\times\frac{4}{7}+3\times(-\frac{3}{7})\)
\(=\frac{9}{7}-\frac{8}{7}+\frac{3}{7}+0-\frac{3}{7}+\frac{8}{7}-\frac{9}{7}=0\)
\(\sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}=(1 - 4)^{2}+(2 - 4)^{2}+(3 - 4)^{2}+(4 - 4)^{2}+(5 - 4)^{2}+(6 - 4)^{2}+(7 - 4)^{2}\)
\(=9 + 4+1+0 + 1+4+9=28\)
Since \(b = 0\), the regression equation is \(y=\bar{y}\) (because \(y=b(x-\bar{x})+\bar{y}\), and \(b = 0\) gives \(y=\bar{y}\))

Answer:

LaTasha is correct. The slope \(b\) of the regression line is \(0\). The regression equation is \(y=\frac{31}{7}\approx4.43\), which means that the value of \(y\) does not change with the change of \(x\) (there is no linear relationship between \(x\) and \(y\) as the slope of the regression line, which measures the strength and direction of the linear correlation, is zero). So, there is no linear correlation between \(x\) and \(y\) in the context of linear regression.