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Chapter 10: Q5BSC (page 468)

In Exercises 5鈥�8, we want to consider the correlation between heights of fathers and mothers and the heights of their sons. Refer to theStatCrunch display and answer the given questions or identify the indicated items.

The display is based on Data Set 5 鈥淔amily Heights鈥� in Appendix B.

Identify the multiple regression equation that expresses the height of a son in terms of the height of his father and mother.

Short Answer

Expert verified

The multiple regression equation is

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

Step by step solution

01

Given information

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

02

State the general equation of multiple regression

The multiple regression equation is

\(\hat y = {b_0} + {b_1}{x_1} + {b_2}{x_2} + ... + {b_n}{x_n}\), where \(\hat y\) is the predicted response variable and \({x_1},{x_2},...,{x_n}\)are the independent variables along with respective measure of coefficients \({b_0},{b_1},...,{b_n}\).

03

Compute the equation of multiple regression

Define the variables as follows:

Father: Height of father

Mother: Height of mother

Son: Height of son

From the output, the estimates of the variables are

\(\begin{array}{l}{b_0} = 17.96657,\\{b_1} = 0.50354896,\\{b_2} = 0.27714316.\end{array}\)

The multiple regression equation for the provided scenario is represented below:

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

Therefore, the multiple regression equation is

\({\rm{Son}} = 18.0 + 0.504\;{\rm{Father}} + 0.277\;{\rm{Mother}}\).

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Most popular questions from this chapter

The following exercises are based on the following sample data consisting of numbers of enrolled students (in thousands) and numbers of burglaries for randomly selected large colleges in a recent year (based on data from the New York Times).

Exercise 1 stated that ris found to be 0.499. Does that value change if the actual enrollment values of 53,000, 28,000, 27,000, 36,000, and 42,000 are used instead of 53, 28, 27, 36, and 42?

Exercises 13鈥�28 use the same data sets as Exercises 13鈥�28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Using the listed duration and interval after times, find the best predicted 鈥渋nterval after鈥� time for an eruption with a duration of 253 seconds. How does it compare to an actual eruption with a duration of 253 seconds and an interval after time of 83 minutes?

The following exercises are based on the following sample data consisting of numbers of enrolled students (in thousands) and numbers of burglaries for randomly selected large colleges in a recent year (based on data from the New York Times).

The sample data result in a linear correlation coefficient of r= 0.499 and the regression equation\(\hat y = 3.83 + 2.39x\). What is the best predicted number of burglaries, given an enrollment of 50 (thousand), and how was it found?

Testing for a Linear Correlation. In Exercises 13鈥�28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

CSI Statistics Police sometimes measure shoe prints at crime scenes so that they can learn something about criminals. Listed below are shoe print lengths, foot lengths, and heights of males (from Data Set 2 鈥淔oot and Height鈥� in Appendix B). Is there sufficient evidence to conclude that there is a linear correlation between shoe print lengths and heights of males? Based on these results, does it appear that police can use a shoe print length to estimate the height of a male?

Shoe print(cm)

29.7

29.7

31.4

31.8

27.6

Foot length(cm)

25.7

25.4

27.9

26.7

25.1

Height (cm)

175.3

177.8

185.4

175.3

172.7

Interpreting the Coefficient of Determination. In Exercises 5鈥�8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Bears r = 0.783 (x = head width of a bear, y = weight of a bear)
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