Heights of husbands and wives. The mean height of American women in their early twenties is about 64.5 inches and the standard deviation is about 2.5 inches. The mean height of men the same age is about 68.5 inches, with standard deviation about 2.7 inches. If the correlation between the heights of husbands and wives is about r = 0.5,a) what is the equation of the regression line of the husband’s height on the wife’s height in young couples?b) Predict the height of the husband of a woman who is 67 inches
a) slope (.5 x 2.7)/ 2.5 = .54 = slope
y hat = ? + .54x
How do you find the y-intercept?
To find the y-intercept of the regression line, we can use the equation of a regression line, which is given by:
y hat = b0 + b1x,
where y hat is the predicted value of the dependent variable (husband's height), b0 is the y-intercept, b1 is the slope (which we have already calculated as 0.54), and x is the independent variable (wife's height).
In this case, we can use the correlation coefficient (r) and the standard deviations of the husband and wife heights to calculate the y-intercept. The equation to find the y-intercept (b0) is given by:
b0 = (mean of husband heights) - b1 * (mean of wife heights),
b0 = (mean of husband heights) - 0.54 * (mean of wife heights).
Given that the mean height of husbands (men) is about 68.5 inches and the mean height of wives (women) is about 64.5 inches, we can substitute these values into the equation:
b0 = 68.5 - 0.54 * 64.5.
Simplifying the equation:
b0 = 68.5 - 0.54 * 64.5
= 68.5 - 34.83
= 33.67.
Therefore, the y-intercept of the regression line of the husband's height on the wife's height is 33.67 inches.
So, the equation of the regression line would be:
y hat = 33.67 + 0.54x.
Now we can proceed to part b of the question.