Question

Husbands and wives The mean height of married American women in their

early 20s is 64.5 inches and the standard deviation is 2.5 inches. The mean height of married men the same age is 68.5 inches with standard deviation 2.7 inches. The correlation between the heights of husbands and wives is about r = 0.5.

a. 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 20s.

b. Suppose that the height of a randomly selected wife was 1 standard deviation below average. Predict the height of her husband without using the least-squares line.

Short answer

Part (a)$\stackrel{⏜}{y}=33.67+0.54x$

Part (b)$67.15$inches.

Step-by-step solution

Part (a) Step 1: Given information

$$\begin{gathered} r=0.5 \\ \overline{x}=64.5 \\ {s}_x=2.5 \\ \overline{y}=68.5 \\ {s}_y=2.7 \end{gathered}$$

Part (a) Step 2: Explanation

The general equation of the regression line is:

$\stackrel{⏜}{y}=a+bx$

The slope of the regression line is:

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

And the intercept is as:

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

Thus, the equation of the regression line is:

$$\begin{gathered} \stackrel{⏜}{y}=a+bx \\ =33.67+0.54x \end{gathered}$$

Part (b) Step 1: Explanation

As a result, by noting that the wife's height is one standard deviation below the mean for women and that the man's expected height must match the correlation coefficient's product and the standard deviation below the mean, we can intuitively arrive at a solution:

$$\begin{gathered} \stackrel{⏜}{y}=\overline{y}+r{s}_y \\ =68.5-0.5(2.7) \\ =67.15 \end{gathered}$$