7.37 Husbands and wives, Part III.

The scatterplot below summarizes husbands’ and wives’ heights in a random sample of 170 married couples in Britain, where both partners’ ages are below 65 years. Summary output of the least squares fit for predicting wife’s height from husband’s height is also provided in the table.

(a) Is there strong evidence that taller men marry taller women? State the hypotheses and include any information used to conduct the test?

Using a one-sided test, the hypotheses are:

  • \(H_o: \beta_1=0\)
  • \(H_A: \beta_1>0\)

The p-value from the table is practically 0, which gives strong evidence in rejecting \(H_o\) in favor \(H_A\). That is, the data do provide convincing evidence that a taller man will marry a taller woman.

(b) Write the equation of the regression line for predicting wife’s height from husband’s height.

\[\widehat{height}_W = 43.5755 + 0.2863 \times \widehat{height}_H\]

(c) Interpret the slope and intercept in the context of the application.

The slope in this context, 0.2863, means that for every inch increase in a husband’s height, his wife’s height is expected to increase by 0.2863“.

The intercept, 43.5755", is the expected height of the wife when the husband’s height is 0. This does not hold any meaning in this context; it is just an adjustment for the linear model.

(d) Given that \(R^2\) = 0.09, what is the correlation of heights in this data set?

Since the slope is positive, the correlation, \(r\), will also be positive. \(r\) is simply the square root of \(R^2\):

\[R^2 = 0.09 \rightarrow r = \sqrt{R^2} = \sqrt{0.09} = 0.3\]

The correlation of 0.3 implies a weak-positive correlation between a wife and husband’s height.

(e) You meet a married man from Britain who is 5’9" (69 inches). What would you predict his wife’s height to be? How reliable is this prediction?

Using the equation derived from (b) and substituting 69" into \(\widehat{height_H}\): \[\widehat{height}_W = 43.5755 + 0.2863 \times 69" = 63.3302"\]

Since \(R^2\) is low, only 9% of the variability in the wife’s height is explained by the husband’s height, the prediction is not very reliable.

(f) You meet another married man from Britain who is 6’7" (79 inches). Would it be wise to use the same linear model to predict his wife’s height? Why or why not?

The height of this man is outside the limits of what this model was based off on, which is approximately \(61" <= height_H <= 75"\), so it would not be wise to use the same model.

Using the least squares equation from (b), it would predict the height of the wife as 66.1932“. At the lower end and upper ends of the height spectrum, there are fewer and fewer”mates" in the pool that would be available to the respective partners to have this equation hold true. This model, at best, should only be used for the heights in the range of height distributions presented, and it should be used with some reservations, due to the low correlation.