91Ó°ÊÓ

Consider the stock market: "percent change in a stock market index in January $\left(X\right)$will assist explain "percent change in a stock market index for the entire year $\left(Y\right)$in this situation. As a result, the explanatory variable is "percent change in a stock market index in January $\left(X\right)$, and the response variable is "percent change in a stock market index for the entire year $\left(Y\right)$.

The following shows the correlation between the percent change in a stock market index over the course of the year and the percent change in a stock market index in January:

$r=0.596$

The least-squares regression line is

$\stackrel{Áåž}{Y}=a+bX$

where

slope

localid="1649409442067" $b=r\frac{S_r}{S_x}$

and Y intercept

$a=\overline{Y}-b\overline{X}$

The slope of the least-squares regression line of percent change in a stock market index during the course of the year on percent change in a stock market index in January is

localid="1650002928470" $$\begin{gathered} b=r\frac{S_r}{S_x} \\ b=0.596×\frac{15.35}{5.36} \\ =1.707 \end{gathered}$$

We use the fact that the least-squares line passes through $(\overline{X},\overline{Y})$to find the intercept, :

localid="1650002934125" $$\begin{gathered} a=\overline{Y}-b\overline{X} \\ =9.07−1.707×1.75 \\ =6.083 \end{gathered}$$

So the equation of the least-squares line is:

$\stackrel{Áåž}{Y}=6.083+1.707X$

Given in the question that the mean change in January is $\overline{\mathit{X}}\mathit{=}\mathit{1}\mathit{.}\mathit{75}\mathit{\%}$.We have to predict the change in the index in a year in which the index rises $1.75\%$in January

According to the information, the least-squares line passes through :

$\overline{\mathit{X}}\mathit{,}\overline{\mathit{Y}}$:

$\overline{\mathit{Y}}\mathit{=}a\mathit{+}b\overline{\mathit{X}}$

Since

$X\mathit{=}\overline{\mathit{X}}$

$\stackrel{\mathit{Áåž}}{\mathit{Y}}\mathit{=}\overline{\mathit{Y}}$

Therefore

$\stackrel{Áåž}{Y}=9.07\%$

We can use a regression line to predict the response for a $\stackrel{Áåž}{Y}$specific value of the explanatory variable $X$. Since we know the values of ,$\overline{X}\&\overline{Y}$we need not to do calculations again.