Chapter 3.2, Problem 47E

(a)

To determine

To find: the equation of the least-squares regression line

Answer to Problem 47E

The equation of the least-squares line is $\widehat{Y}=33.67+0.54X$

Explanation of Solution

Given:

	Mean	Standard deviation
Wife’s height $\left(X\right)$	$\overline{X}=64.5$	$S_X=2.5$
Husband’s height $\left(Y\right)$	$\overline{Y}=68.5$	$S_Y=2.7$

The correlation between the heights of husbands and wives is given as:

  $r=0.5$

Calculation:

The least-squares regression line is the line:

  $\widehat{Y}=a+bX$

With slope$\widehat{Y}=a+bX$

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

The least-squares regression line of husband’s height on wife’s height has slope:

  $\begin{array}{l}b=0.5\times \frac{2.7}{2.5}\\ =0.54\end{array}$

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

  $\begin{array}{l}\overline{Y}=a+b\overline{X}\\ a=68.5-0.54\times 64.5\\ =33.67\end{array}$

Conclusion:

So the equation of the least-squares line is:

  $\widehat{Y}=33.67+0.54X$

(b)

To determine

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

Answer to Problem 47E

A regression line is used to predict the response $\widehat{Y}$ for a specific value of the explanatory variable $X$ .

Explanation of Solution

Calculation:

The predicted height of the husband of a woman who is 67 inches tall is:

  $\begin{array}{l}\widehat{Y}=33.67+0.54X\\ =33.67+0.54\times 67\\ =69.85\\ \widehat{Y}=69.85inches\end{array}$

A regression line is used to predict the response $\widehat{Y}$ for a specific value of the explanatory variable $X$ .

Conclusion:

Hence, a regression line is used to predict the response $\widehat{Y}$ for a specific value of the explanatory variable $X$ .