Question

In Exercises 5–8, we want to consider the correlation between heights of fathers and mothers and the heights of their sons. Refer to the

StatCrunch display and answer the given questions or identify the indicated items.

The display is based on Data Set 5 “Family Heights” in Appendix B.

A son will be born to a father who is 70 in. tall and a mother who is 60 in. tall. Use the multiple regression equation to predict the height of the son. Is the result likely to be a good predicted value? Why or why not?

Short answer

Thepredictedheight of the sonborn to a father who is 70 in. tall and a mother who is 60 in. tallis 69.8 in.

The predicted value is not ideal as the model is not a good fit due to relatively lower values of R-squared measures.

Step-by-step solution

Given information

The analysis of variance table for multiple regression model is provided.

State the general equation of multiple regression

The multiple regression equation is

\(\hat y = {b_0} + {b_1}{x_1} + {b_2}{x_2} + ... + {b_n}{x_n}\).

Obtain the equation of multiple regression

The multiple regression equation for the provided scenario is represented:

\(\begin{array}{c}Son = {b_0} + {b_1}\;Father + {b_2}\;Mother\\ = 17.9666 + 0.504\;Father + 0.277\;Mother\end{array}\)

Here, each coefficient is obtained from the output table.

Therefore,the multiple regression equation for predicting the height of the son is

\(Son = 18 + 0.504\;Father + 0.277\;Mother\).

Predict the height of the son

The predictedheight of the sonborn to a father who is 70 in. tall and a mother who is 60 in. tallis

\(\begin{array}{c}Son = 18 + 0.504\;Father + 0.277\;Mother\\ = 18 + \left( {0.504 \times 70} \right) + \left( {0.277 \times 60} \right)\\ = 69.84.\end{array}\)

State if the result is likely to be a good predicted value

Based on the output, the P-valuein the last column ofthevariance table for the multiple regression model is low. It is less than 0.0001, indicating a significant model.

But the coefficient of determination and the adjusted coefficient of determination at 0.3249 and 0.3552, respectively, are not high.

Therefore,the multiple regression equation fits the sample data, but it is not a good fit.

Thus, the value is not a good predicted value.