Chapter 3.2, Problem 63E

(a)

To determine

To find the equation of the least squares regression line for predicting a husband’s height from his wife’s height for married couples in their early $20$ ’s.

Answer to Problem 63E

  $\widehat{y}=33.67+0.54x$ .

Explanation of Solution

It is given that,

  $\begin{array}{l}r=0.5\\ \overline{x}=64.5\\ s_x=2.5\\ \overline{y}=68.5\\ s_y=2.7\end{array}$

The general equation of the regression line is:

  $\widehat{y}=a+bx$

The slope of the regression line is:

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

And the intercept is as:

  $\begin{array}{l}a=\overline{y}-b\overline{x}\\ =68.5-0.54(64.5)\\ =33.67\end{array}$

Thus, the equation of the regression line is:

  $\begin{array}{l}\widehat{y}=a+bx\\ =33.67+0.54x\end{array}$

(b)

To determine

To predict the height of her husband without using the least square line.

Answer to Problem 63E

  $67.15$ inches.

Explanation of Solution

It is given that,

  $\begin{array}{l}r=0.5\\ \overline{x}=64.5\\ s_x=2.5\\ \overline{y}=68.5\\ s_y=2.7\end{array}$

Thus, we can intuitively obtain a solution by realizing that the wife’s height is one standard deviation below the mean for women and thus the predicted height of the man has to be the product of the correlation coefficient and the standard deviation below its mean:

  $\begin{array}{l}\widehat{y}=\overline{y}+r{s}_y\\ =68.5-0.5(2.7)\\ =67.15\end{array}$