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.

x	1	2	3	4	5	6	7
y	4	5	4	5	4	5	4

Explanation:

Step1: Calculate the means of x and y

$\bar{x}=\frac{1 + 2+3+4+5+6+7}{7}=\frac{28}{7}=4$
$\bar{y}=\frac{4 + 5+4+5+4+5+4}{7}=\frac{31}{7}\approx4.43$

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

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$
$\sum_{i = 1}^{7}(x_i - 4)(y_i-4.43)=(1 - 4)(4 - 4.43)+(2 - 4)(5 - 4.43)+(3 - 4)(4 - 4.43)+(4 - 4)(5 - 4.43)+(5 - 4)(4 - 4.43)+(6 - 4)(5 - 4.43)+(7 - 4)(4 - 4.43)$
$=(-3)(-0.43)+(-2)(0.57)+(-1)(-0.43)+(0)(0.57)+(1)(-0.43)+(2)(0.57)+(3)(-0.43)$
$=1.29-1.14 + 0.43+0-0.43+1.14-1.29 = 0$
$\sum_{i = 1}^{7}(x_i - 4)^2=(1 - 4)^2+(2 - 4)^2+(3 - 4)^2+(4 - 4)^2+(5 - 4)^2+(6 - 4)^2+(7 - 4)^2$
$=9 + 4+1+0+1+4+9 = 28$
Since $b = 0$, the regression equation is of the form $y=a+0x=a$.

Step3: Determine the correlation

Since the slope of the regression line is 0, there is no linear - relationship (correlation) between $x$ and $y$. So LaTasha is correct.

Answer:

LaTasha is correct. There is no correlation between x and y as the slope of the regression line is 0.