Question

The stock market Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year. Take the explanatory variable x to be the percent change in a stock market index in January and the response variable y to be the change in the index for the entire year. We expect a positive correlation between x and y because the change during January contributes to the full year’s change. Calculation from data for an 18-year period gives x = $$\begin{gathered} \overline{X}=1.75\%s_x=5.36\%\overline{y}=9.07\% \\ {s}_y=15.35\%r=0.596 \end{gathered}$$

(a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work.

(b) The mean change in January is $x=1.75\%$. Use your regression line to predict the change in the index in a year in which the index rises$1.75\%$ in January. Why could you have given this result (up to roundoff error) without doing the calculation?

Short answer

(a) The equation of the least squares line is $\overbrace{Y}=6.083+1.707X$

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

Step-by-step solution

Part (a) Step 1: Given Information 

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 $\left(Y\right)$.

Part (a) Step 2: Explanation

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

$\overbrace{Y}=a+bX$

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

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

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

So the equation of the least-squares line is:

$\overbrace{Y}=6.083+1.707X$

Part (b) Step 1:  Given Information

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

Part (b) Step 2: Explanation 

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

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

$\overbrace{\mathit{Y}}\mathit{=}\overline{\mathit{Y}}$

Therefore

$\overbrace{Y}=9.07\%$

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