Husbands and wives The mean height of American women in their early twenties is $64.5$inches and the standard deviation is $2.5$inches. The mean height of men the same age is $68.5$inches, with a standard deviation of $2.7$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 the husband’s height from the wife’s height. Show your work.

(b) Use your regression line to predict the height of the husband of a woman who is $67$inches tall. Explain why you could have given this result without doing the calculation

It is given in the question that,

According to the table, the heights of husbands and wives are correlated as:$r=0.5$

In regression analysis, the least-squares line is:

$\stackrel{\wedge }{Y}=a+bX$with slope $\stackrel{\wedge }{Y}=a+bX$and$Y$intercept $a$= $\stackrel{-}{Y}-b\stackrel{-}{X}$

Based on the least-squares regression line plotting husband's height against wife's height, the slope is as follows:

localid="1649943536880" $$\begin{gathered} b=0.5\times \frac{2.7}{2.5} \\ =0.54 \end{gathered}$$

Since the least-squares line passes through$\left(\stackrel{-}{X,}\stackrel{-}{Y}\right)$we use this information to find the intercept.

localid="1649943542257" $$\begin{gathered} \stackrel{-}{Y}=a+b\stackrel{-}{X} \\ a=68.5-0.54\times 64.5 \\ =33.67 \end{gathered}$$

It is given in the question that, The correlation between the heights of husbands and wives is about r = $0.5$.

Use your regression line to predict the height of the husband of a woman who is $67$inches tall.

Based on a woman who stands$67$inches tall, the height of her husband is,

localid="1650003095350" $$\begin{gathered} \stackrel{\wedge }{Y}=33.67+0.54X \\ =33.67+0.54\times 67 \\ =69.85 \end{gathered}$$

$\stackrel{\wedge }{Y}=69.85inches$

For a specific value of the explanatory variable localid="1649943566363" $X$, a regression line is used to predict the responselocalid="1649943572450" $Y$.