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Least-Squares Method and Coefficient of Determination

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The director of graduate studies at a large college of business would like to predict the grade point average (gpa) of students in an MBA program based on Graduate Management admission test (GMAT) scores. A sample of 20 students who had completed 2 years in the program is selected. The results are as follows:

Observation GMAT Score GPA
1 688 3.72
2 647 3.44
3 652 3.21
4 608 3.29
5 680 3.91
6 617 3.28
7 557 3.02
8 599 3.13
9 616 3.45
10 594 3.33
11 567 3.07
12 542 2.86
13 551 2.91
14 573 2.79
15 536 3.00
16 639 3.55
17 619 3.47
18 694 3.60
19 718 3.88
20 759 3.76

(Hint: First, determine which are the independent and dependendent variables)
a.Construct a scatter plot and, assuming a linear relationship, use the least-squares method to compute the regression coeffients b(0) and b(1)

b. Interpret the meaning of the y-intercept, b(0), and the slope, b(1), in this problem.

c. Use the prediction line developed in (a) to predict the GPA for a student a GMAT score of 600

d. Determine the coefficient of determination, r^2, and interpret its meaning in this problem

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Solution Summary

This solutions contains a graphical display of the data to determine the regression equation and the coefficient of determination. It also interprets the y-intercept and predicts the GPA for a GMAT score of 600.

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Questions and Problems

1. Regression analysis is a statistical procedure for developing a mathematical equation that describes how:
a. one independent and one or more dependent variables are related
b. several independent and several dependent variables are related
c. one dependent and one or more independent variables are related.
d. None of the above answers is correct.
2. In a simple regression analysis (where Y is a dependent and X an independent variable), if the Y intercept is positive, then:
a. there is a positive correlation between X and Y
b. there is a negative correlation between X and Y
c. if X is increased, Y must also increase
d. if Y is increased, X must also increase
e. None of the above answers is correct.
3. In regression analysis, the variable that is being predicted is the:
a. dependent variable
b. independent variable
c. intervening variable
d. None of the above answers is correct.
4. A procedure used for finding the equation of a straight line that provides the best approximation for the relationship between the independent and dependent variables is the:
a. correlation analysis
b. mean squares method
c. least squares method
d. most squares method
e. None of the above answers is correct.

5. If the coefficient of determination is equal to 1, then the coefficient of correlation:
a. must also be equal to 1
b. can be either -1 or +1
c. can be any value between -1 to +1
d. must be -1
e. None of the above answers is correct.
6. In a regression analysis, the variable that is being predicted:
a. must have the same units as the variable doing the predicting
b. is the independent variable
c. is the dependent variable
d. usually is denoted by x
e. None of the above answers is correct.

7. Shown below is a partial computer output from a regression analysis.
Predictor Coefficient Stdev
Constant 10.00 2.00
X1 -2.00 1.50
X2 6.00 2.00
X3 -4.00 1.00
Analysis of Variance
Source of Degrees Sum of Mean
Variation of Freedom Squares Square F
Regression 60
Error
Total 19 140
a. Use the above results and write the regression equation.
b. Compute the coefficient of determination and fully interpret its meaning.
c. At  = 0.05, test to see if there is a relation between X1 and Y.
d. At  = 0.05, test to see if there is a relation between X3 and Y.
e. Is the regression model significant? Perform an F test and let  = 0.05.

8. The Very Fresh Juice Company has developed a regression model relating sales (Y in $10,000s) with four independent variables. The four independent variables are price per unit (X1, in dollars), competitor's price (X2, in dollars), advertising (X3, in $1,000s) and type of container used (X4) (1 = Cans and 0 = Bottles). Part of the regression results are shown below.
Analysis of Variance
Source of Degrees of Sum of
Variation Freedom Squares
Regression 4 283,940.60
Error (Residuals) 18 621,735.14
a. Compute the coefficient of determination and fully interpret its meaning.
b. Is the regression model significant? Explain what your answer implies. Let  = 0.05.
c. What has been the sample size for this analysis?

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