Concept explainers
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
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
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?
1.
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.
2.
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.
3.
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.
4.
To explain: The numbers 4540 and 1290 in the regression equation.
Explanation of Solution
Justification:
From the given information, the regression equation is
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.
5.
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
Substitute 8 for
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.
6.
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.
7.
To compute: The value of
Answer to Problem 1AC
The value of
Explanation of Solution
Calculation:
The value of
Thus, the value of
8.
To find: The adjusted
Answer to Problem 1AC
The adjusted
Explanation of Solution
Calculation:
The formula for finding adjusted
Substitute 0.857 for
Thus, the adjusted
9.
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
10.
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.
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