Question

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 \(y\)

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 \(b_1\) of the regression line

The formula for \(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\(\sum_{i = 1}^{7}(x_i - 4)(y_i-4.43)\):
\((1 - 4)(4 - 4.43)+(2 - 4)(5 - 4.43)+(3 - 4)(4 - 4.43)+(4 - 4)(5 - 4.43)+(5 - 4)(4 - 4.43)+(6 - 4)(5 - 4.43)+(7 - 4)(4 - 4.43)\)
\(=(- 3)(-0.43)+(-2)(0.57)+(-1)(-0.43)+(0)(0.57)+(1)(-0.43)+(2)(0.57)+(3)(-0.43)\)
\(=1.29-1.14 + 0.43+0-0.43+1.14-1.29=0\)
\(\sum_{i=1}^{7}(x_i - 4)^2=(1 - 4)^2+(2 - 4)^2+(3 - 4)^2+(4 - 4)^2+(5 - 4)^2+(6 - 4)^2+(7 - 4)^2\)
\(=9 + 4+1+0+1+4+9 = 28\)
Since \(b_1=\frac{0}{28}=0\), the regression equation is \(y=b_0 + 0x=b_0\) (where \(b_0=\bar{y}\approx4.43\)).

Step3: Analyze the correlation

A slope of \(0\) in the regression equation indicates that there is no linear - relationship between \(x\) and \(y\).

Answer:

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