Chapter 10.4, Problem 1AC

Applying the Concepts 10–4

More Math Means More Money

In a study to determine a person’s yearly income 10 years after high school, it was found that the two biggest predictors are number of math and science courses taken and number of hours worked per week during a person’s senior year of high school. The multiple regression equation generated from a sample of 20 individuals is

y′ = 6000 + 4540x1 + 1290x₂

Let x1 represent the number of math and science courses taken and x₂ represent hours worked during senior year. The correlation between income and math and science courses is 0.63. The correlation between income and hours worked is 0.84, and the correlation between math and science courses and hours worked is 0.31. Use this information to answer the following questions.

1. What is the dependent variable?

2. What are the independent variables?

3. What are the multiple regression assumptions?

4. Explain what 4540 and 1290 in the equation tell us.

5. What is the predicted income if a person took 8 math and science classes and worked 20 hours per week during her or his senior year in high school?

6. What does a multiple correlation coefficient of 0.926 mean?

7. Compute R².

8. Compute the adjusted R².

9. Would the equation be considered a good predictor of income?

10. What are your conclusions about the relationship among courses taken, hours worked, and yearly income?

To determine

To find: The dependent variable.

Answer to Problem 1AC

The dependent variable is a person’s yearly income 10 years after high school.

Explanation of Solution

Given info:

The data shows that the correlation between income and math and science courses is 0.63. The correlation between income and hours worked is 0.84, and the correlation between math and science courses and hours worked is 0.31.

Justification:

Here, a person’s yearly income 10 years after school is obtained by using the predictor’s number of math and science courses taken and number of hours worked per week during a person’s senior year of high school.

Thus, the dependent variable is a person’s yearly income 10 years after high school.

To determine

To find: The independent variables.

Answer to Problem 1AC

The independent variables are number of math and science courses taken and number of hours worked per week during a person’s senior year of high school.

Explanation of Solution

Justification:

Here, the predictor’s number of math and science courses taken and number of hours worked per week during a person’s senior year of high school are used to predict a person’s yearly income 10 years after school is obtained by using

Thus, the independent variables are number of math and science courses taken and number of hours worked per week during a person’s senior year of high school.

To determine

To write: The assumptions for the multiple regression.

Answer to Problem 1AC

The assumption is that the independent variables number of math and science courses taken and number of hours worked per week during a person’s senior year of high school are not correlated.

Explanation of Solution

Justification:

The main assumption of the multiple regression assumption is that correlation between number of math and science courses taken and number of hours worked per week during a person’s senior year of high school is less.

To determine

To explain: The numbers 4540 and 1290 in the regression equation.

Explanation of Solution

Justification:

From the given information, the regression equation is $y^{\text{'}}=6000+4540x_1+1290x_2$.

Interpretation of 4540:

It can said that by keeping the number of hours as constant and one unit increase in the number of math and science courses, a person’s yearly income 10 years after high school increases by $4,540.

Interpretation of 1290:

It can said that by keeping the number of math and science courses as constant and one unit increase in the number of hours, a person’s yearly income 10 years after high school increases by $1,290.

To determine

To find: The predicted income if a person took 8 math and science classes and worked 20 hours per week during her of his senior year in high school.

Answer to Problem 1AC

The predicted income if a person took 8 math and science classes and worked 20 hours per week during her of his senior year in high school is $68,120.

Explanation of Solution

Calculation:

From the given information, the regression equation is $y^{\text{'}}=6000+4540x_1+1290x_2$

Substitute 8 for $x_1$ and 20 for $x_2$

$\begin{array}{c}{y}^{\text{'}}=6000+4540x_1+1290x_2\\ =6000+4540\left(8\right)+1290\left(20\right)\\ =6000+36320+25800\\ =68,120\end{array}$

Thus, the predicted income if a person took 8 math and science classes and worked 20 hours per week during her of his senior year in high school is $68,120.

To determine

To explain: The meaning of a multiple correlation coefficient of 0.926.

Answer to Problem 1AC

There is strong positive correlation between the dependent variable and independent variables.

Explanation of Solution

Justification:

The multiple correlation coefficient gives the correlation between independent variables. Here, the multiple correlation coefficient is 0.926. That is, there is strong positive correlation between the dependent variable and independent variables.

To determine

To compute: The value of $R^2$.

Answer to Problem 1AC

The value of $R^2$ is 0.857.

Explanation of Solution

Calculation:

The value of $R^2$ is,

$\begin{array}{c}{R}^2={\left(\text{Multiple}\text{correlation}\text{coefficient}\right)}^2\\ ={0.926}^2\\ =0.857\end{array}$

Thus, the value of $R^2$ is 0.857.

To determine

To find: The adjusted $R^2$.

Answer to Problem 1AC

The adjusted $R^2$ is 0.840.

Explanation of Solution

Calculation:

The formula for finding adjusted $R^2$ is $R_{\text{adj}}^2=1-\frac{\left(1-R^2\right)\left(n-1\right)}{n-k-1}$

Substitute 0.857 for $R^2$, 20 for n and 2 for k

$\begin{array}{c}{R}_{\text{adj}}^2=1-\left[\frac{\left(1-R^2\right)\left(n-1\right)}{n-k-1}\right]\\ =1-\left[\frac{\left(1-0.857\right)\left(20-1\right)}{20-2-1}\right]\\ =1-\left[\frac{\left(1-0.857\right)\left(19\right)}{17}\right]\\ =1-\frac{2.717}{17}\end{array}$

$\begin{array}{c}=1-0.1598\\ =0.840\end{array}$

Thus, the adjusted $R^2$ is 0.840.

To determine

To explain: Whether the equation be considered a good predictor of income.

Answer to Problem 1AC

The equation be considered a good predictor in income.

Explanation of Solution

Justification:

From the part (7), the value of $R^2$ is 0.857. That is, 85.7% of the variation in dependent variable is explained by the two independent variables. The most of the variation is explained. Thus, the equation be considered a good predictor in income.

To determine

To find: The conclusions about the relationship among courses taken, hour’s worked and yearly income.

Answer to Problem 1AC

It can be concluded that the person’s yearly income increases with the increase in the number of math and science courses taken and hours worked during senior year.

Explanation of Solution

Justification:

As the variables number of math and science courses taken and hours worked during senior year, the person’s yearly income increases. Thus, it can be concluded that the person’s yearly income increases with the increase in the number of math and science courses taken and hours worked during senior year.