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 $r = 0.5$. 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. the predicted value for the husband is inches. (round to 2 decimal places)

Explanation:

Step1: Recall the concept of correlation

The correlation coefficient $r$ gives the relationship between two variables. When the wife's height is 1 standard - deviation below average, we use the formula for prediction based on correlation. The predicted change in the husband's height (in standard - deviation units) is $r$ times the change in the wife's height (in standard - deviation units).

Step2: Calculate the predicted change in the husband's height in standard - deviation units

The wife's height is 1 standard - deviation below average, so the change in the wife's height in standard - deviation units is $- 1$. Given $r = 0.5$, the predicted change in the husband's height in standard - deviation units is $r\times(-1)=0.5\times(-1)=-0.5$.

Step3: Calculate the predicted height of the husband

The mean height of husbands is $\mu_{husband}=68.5$ inches and the standard deviation of husbands' heights is $\sigma_{husband}=2.7$ inches. The predicted height of the husband $y$ is given by $y=\mu_{husband}+(-0.5)\times\sigma_{husband}$.
\[y = 68.5+(-0.5)\times2.7=68.5 - 1.35=67.15\]

Answer:

$67.15$