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.

Question

Accepted Answer

# 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
$$
\begin{align*}
b&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\
\sum_{i = 1}^{7}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(1 - 4)(4-\frac{31}{7})+(2 - 4)(5-\frac{31}{7})+(3 - 4)(4-\frac{31}{7})+(4 - 4)(5-\frac{31}{7})+(5 - 4)(4-\frac{31}{7})+(6 - 4)(5-\frac{31}{7})+(7 - 4)(4-\frac{31}{7})\
&=(- 3)\times(-\frac{3}{7})+(-2)\times(\frac{4}{7})+(-1)\times(-\frac{3}{7})+0\times(\frac{4}{7})+1\times(-\frac{3}{7})+2\times(\frac{4}{7})+3\times(-\frac{3}{7})\
&=\frac{9}{7}-\frac{8}{7}+\frac{3}{7}+0-\frac{3}{7}+\frac{8}{7}-\frac{9}{7}\
&=0
\end{align*}
$$
$$
\begin{align*}
\sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}&=(1 - 4)^{2}+(2 - 4)^{2}+(3 - 4)^{2}+(4 - 4)^{2}+(5 - 4)^{2}+(6 - 4)^{2}+(7 - 4)^{2}\
&=9 + 4+1+0+1+4+9\
&=28
\end{align*}
$$
Since $b = 0$, the regression equation is $y=a$. Using the formula $a=\bar{y}-b\bar{x}$, and since $b = 0$, $a=\bar{y}\approx4.43$. The regression equation is $y = 4.43$ (a horizontal line).

# Answer:
LaTasha is correct. Since the slope $b = 0$ in the regression equation $y=a$ (a horizontal line), there is no linear correlation between $x$ and $y$.

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QUESTION IMAGE

Question

Explanation:

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

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

Answer: