Chapter 7 Problem 37

Is there strong Evidence that taller men marry taller Women?

For Part A, we would need to set up a Hypothesis test to determine whether or not \( \beta_1 \) is greater than 0. Our Hypothesis test would be:

\( H_0: \beta_1 = 0 \) The linear regression has a slope of 0
\( H_A: \beta_1 > 0 \) The slope is greater than 0.

We see from the output of the least square equation, that the p-value is incredibly small (approaching 0). This support our alternate hypothesis, that the slope is greater than 0, and is indicative that there is convincing evidence that there is a positive correlation for husbands and wives' heights.

Write the equation of the regression line:

Height_Wife = 43.5755 + 0.2863 * Height_Husband

Interpret the slope and intercept in the context of this application:

The slope for this particular equation is 0.2863, which indicates that for each inch a husband's height increases, a wife's height will only be .2863 inches taller.

The Intercept on the other hand is 43.5755. This would indicate that Men who are 0 inches tall would on average have a wife who is 43 inches. This obviously, is meaningless.

Given that \( R^2 \) = 0.09 What is the correlation of heights?

r = \( \sqrt(0.09) \) = 0.30

We know from the regression model that the slope is positive, so we know that r would also be positive. We would solve for r:

Predict a husband's wife height if they are 69 and 79 inches?

Wife_Height = 43.5755 + .2863 * 69 = 63.3302
Wife_Height = 43.5755 + .2863 * 79 = 66.1932

Since \( R^2 \) is quite low it would not be wise to use this formula as a predictor of height. Though there is strong evidence that there is a positively correlated, the regression line we used as a predictor is not a good “fit”.