Question

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. find the equation of the least - squares regression line for predicting a husbands height from his wifes height for married couples in their early 20s. the equation for predicting y = husbands height from x = wifes height is \\(\hat{y}=27.51 + 0.54x\\). the equation for predicting y = husbands height from x = wifes height is \\(\hat{y}=70.82 + 2.16x\\). the equation for predicting y = husbands height from x = wifes height is \\(\hat{y}=38.64 + 0.463x\\). the equation for predicting y = husbands height from x = wifes height is \\(\hat{y}=33.67 + 0.54x\\). the equation for predicting y = husbands height from x = wifes height is \\(\hat{y}=83.46 + 2.16x\\).

Explanation:

Step1: Recall the formula for the slope ($b_1$) of the least - squares regression line

The formula for the slope is $b_1 = r\frac{s_y}{s_x}$, where $r$ is the correlation coefficient, $s_y$ is the standard deviation of the response variable, and $s_x$ is the standard deviation of the explanatory variable. Here, $r = 0.5$, $s_y=2.7$ (standard deviation of men's heights) and $s_x = 2.5$ (standard deviation of women's heights). So, $b_1=0.5\times\frac{2.7}{2.5}=0.54$.

Step2: Recall the formula for the y - intercept ($b_0$) of the least - squares regression line

The formula for the y - intercept is $b_0=\bar{y}-b_1\bar{x}$, where $\bar{y}$ is the mean of the response variable and $\bar{x}$ is the mean of the explanatory variable. $\bar{y} = 68.5$ (mean height of men) and $\bar{x}=64.5$ (mean height of women). Then $b_0=68.5 - 0.54\times64.5=68.5- 34.83=33.67$.

Step3: Write the equation of the least - squares regression line

The equation of the least - squares regression line is $\hat{y}=b_0 + b_1x$. Substituting $b_0 = 33.67$ and $b_1 = 0.54$, we get $\hat{y}=33.67+0.54x$, where $y$ is the husband's height and $x$ is the wife's height.

Answer:

The equation for predicting $y =$ husband's height from $x =$ wife's height is $\hat{y}=33.67 + 0.54x$.