Problem 8.33

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

\( {H}_{0} \): There is no relationship between the heights of husband and wife.
\( {H}_{A} \): There us a relationship between the heights of husband and wife.

The p-value, as reported in the table, is much smaller than 0.05, so we reject \( {H}_{0} \). The data provide convincing evidence that wives' and husbands' heights are positively correlated.

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

\( \hat{{height}_{wife}}=43.5755+0.2863\times{height}_{husband} \)

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

Slope: For each additional inch in husband's height, the average wife's height is expected to be an additional 0.2863 inches on average.

Intercept: Men who are 0 inches tall are expected to have wives 43.5755 inches tall. The intercept here is meaningless.

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

\( Correlation=\sqrt{R^2} \)

\( R=0.30 \)

( 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?

\( \hat{{height}_{wife}}=43.5755+0.2863\times{height}_{husband} \)

\( \hat{{height}_{wife}}=43.5755+0.2863\times69 \)

\( \hat{{height}_{wife}}\approx63\space inches \)

This is not very reliable since \( R^2 \) is small.

( 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?

You should not use the same linear model since 79 inches would be an outlier and exrapolation is bad.