Question

5.34 husbands and wives. the mean height of american women in their twenties is about 64.3 inches, and the standard deviation is about 2.7 inches. the mean height of men of the same age is about 69.9 inches, with standard deviation about 3.1 inches. suppose that the correlation between the heights of husbands and wives is about r = 0.5. a. what are the slope and intercept of the regression line of the husbands height on the wifes height in young couples? interpret the slope in the context of the problem. b. draw a graph of this regression line for heights of wives between 56 and 72 inches. predict the height of the husband of a woman who is 67 inches tall and plot the wifes height and predicted husbands height on your graph. c. you dont expect this prediction for a single couple to be very accurate. why not?

Explanation:

Step1: Calculate the slope

The formula for the slope ($b_1$) of the regression line is $b_1 = r\frac{s_x}{s_y}$, where $r$ is the correlation coefficient, $s_x$ is the standard - deviation of the response variable (husband's height), and $s_y$ is the standard - deviation of the predictor variable (wife's height). Given $r = 0.5$, $s_x=3.1$ inches, and $s_y = 2.7$ inches. Then $b_1=0.5\times\frac{3.1}{2.7}=\frac{1.55}{2.7}\approx0.574$.

Step2: Calculate the intercept

The formula for the intercept ($b_0$) of the regression line is $b_0=\bar{x}-b_1\bar{y}$, where $\bar{x}$ is the mean of the response variable (husband's height) and $\bar{y}$ is the mean of the predictor variable (wife's height). Given $\bar{x}=69.9$ inches and $\bar{y}=64.3$ inches, and $b_1\approx0.574$. Then $b_0 = 69.9-0.574\times64.3=69.9 - 36.9082=32.9918\approx33.0$.

Step3: Interpret the slope

The slope $b_1\approx0.574$ means that for every one - inch increase in the wife's height, the husband's height is expected to increase by approximately $0.574$ inches.

Step4: Predict husband's height

The regression equation is $\hat{x}=b_0 + b_1y=33.0+0.574y$. When $y = 67$ inches, $\hat{x}=33.0+0.574\times67=33.0 + 38.458=71.458\approx71.5$ inches.

Step5: Explain prediction accuracy

The prediction for a single couple is not very accurate because the regression line gives an average relationship. There is variability around the regression line. The correlation coefficient $r = 0.5$ indicates that only $r^{2}=0.25$ or 25% of the variation in husband's height can be explained by the wife's height. There are other factors (e.g., genetic factors not related to the wife's height, environmental factors) that affect the husband's height.

Answer:

a. Slope $b_1\approx0.574$, intercept $b_0\approx33.0$. The slope means for every 1 - inch increase in wife's height, husband's height is expected to increase by about $0.574$ inches.
b. The regression line is $\hat{x}=33.0 + 0.574y$. When $y = 67$ inches, $\hat{x}\approx71.5$ inches. (Graphing requires plotting points for $y$ values between 56 and 72 using the regression equation).
c. Only 25% of the variation in husband's height is explained by wife's height due to $r = 0.5$ ($r^{2}=0.25$), and there are other factors affecting husband's height.