Question

Husbands and wives The mean height of American women in their early twenties is $64.5$inches and the standard deviation is $2.5$inches. The mean height of men the same age is $68.5$inches, with standard deviation$2.7$inches. The correlation between the heights of husbands and wives is about $r=0.5$

(a) Find $r^2$and interpret this value in context.

(b) For these data, $s=1.2$. Explain what this value mean

Short answer

(a) $r^2=0.25$$r^2=0.25$

(b) $s=1.2$shows too much prediction error.

Step-by-step solution

Part (a) Step 1:  Given Information

Given in the question that The correlation between the heights of husbands and wives is about $r=0.5$we have to find out $r^2$

Part (a) Step 2: Explanation 

For these data:

$r^2={(0.5)}^2=0.25$

In regression, $r^2$is a statistic that provides information on a model's goodness of fit. A statistical measure of how well the regression line approximates the real data points is the $r^2$coefficient of determination. An $r^2$of $1.0$indicates that the regression line fully fits the data, but its value is $0.25$, indicating that the regression line does not precisely fit the data, and that the straight-line relationship between $Y$and $X$accounts for around $25\%$of the variation in $Y$.

Part (b) Step 1:  Given Information

Given in the problem that $s=1.2$we have to explain the what this value means.

Part (b) Step 2:  Explanation

For these data the standard deviation of the residuals is $s=1.2$

The standard deviation of the residuals $s$measures the average size of the prediction errors (residuals) when using the regression line. $s=1.2$shows too much prediction error.