Chapter 3.2, Problem 48E

(a)

To determine

To find: the equation of the least-squares line

Answer to Problem 48E

The equation of the least-squares line is $\widehat{Y}=6.083+1.707X$

Explanation of Solution

Given:

	Mean	Standard deviation
Percent change in a stock market index in January $\left(X\right)$	$\overline{X}=1.75\%$	$S_X=5.36\%$
Percent change in a stock market index for the entire year $\left(Y\right)$	$\overline{Y}=9.07\%$	$S_Y=15.35\%$

  $r=0.596$

Calculation:

The least-squares regression line is:

  $\widehat{Y}=a+bX$

With slope$b=r\frac{S_Y}{S_X}$

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

The least-squares regression line of percent change in a stock market index for the entire year on percent change in a stock market index for the January has slope is$\begin{array}{l}b=r\frac{S_Y}{S_X}\\ b=0.596\times \frac{15.35}{5.36}\\ =1.707\end{array}$

To find the intercept, we use the fact that the least-squares line passes through $\left(\overline{X},\overline{Y}\right)$ :

  $\begin{array}{l}a=\overline{Y}-b\overline{X}\\ =9.07-1.707\times 1.75\\ =6.083\end{array}$ g

So the equation of the least-squares line is:

  $\widehat{Y}=6.083+1.707X$

Conclusion:

Hence, the equation of the least-squares line is $\widehat{Y}=6.083+1.707X$

(b)

To determine

To explain:why the result is obtained without doing the calculation.

Answer to Problem 48E

A regression line is used to predict the response $\widehat{Y}$ for a specific value of the explanatory variable $X$ . Since we know the values of $\overline{X}\&\overline{Y}$ , we need not to do calculation again.

Explanation of Solution

Calculation:

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

  $\widehat{Y}=a+bX$

Since$\begin{array}{l}X=\overline{X}\\ Y=\overline{Y}\end{array}$

Therefore$\widehat{Y}=9.07\%$

A regression line is used to predict the response $\widehat{Y}$ for a specific value of the explanatory variable $X$ .Since we know the values of $\overline{X}\&\overline{Y}$ , we need not to do calculation again.

Conclusion:

Hence, a regression line is used to predict the response $\widehat{Y}$ for a specific value of the explanatory variable $X$ .Since we know the values of $\overline{X}\&\overline{Y}$ , we need not to do calculation again.