# Linear Regression

The director of graduate studies at a large college of business would like to be able to predict the grade point index (GPI) of students in an MBA program based on Graduate Management Aptitude Test (GMAT) score. A sample of 20 students who had completed 2 years in the program is selected; the results are as follows:

[see the attached file for the table]

a. Plot a scatter diagram [Using Excel] and, assuming a linear relationship, use the least-squares method to find the regression coefficients b0 & b1.

b. Interpret the meaning of the Y intercept b0 and the slope b1 in this problem.

c. Use the regression equation developed in (a) to predict the GPI for a student with a GMAT score of 600.

d. Determine the standard error of the estimate.

e. Determine the coefficient of determination r² and interpret its meaning in this problem.

f. Determine the coefficient of correlation r.

g. Perform a residual analysis on your results and determine the adequacy of the fit of the model.

h. At the 0.05 level of significance, is there evidence of a linear relationship between GMAT score and GPI?

i. Set up a 95% confidence interval estimate for the average GPI of students with a GMAT score of 600.

j. Set up a 95% prediction interval for a particular student with a GMAT score of 600.

k. Set up a 95% confidence interval estimate of the population slope.

l. Suppose the GPIs of the 19th and 20th students were incorrectly entered. The GPI for student 19 should be 3.76, and the GPI for student 20 should be 3.88. Repeat (a) - (k) and compare the results with your original results.

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#### Solution Preview

Q. 13

The director of graduate studies at a large college of business would like to be able to predict the grade point index (GPI) of students in an MBA program based on Graduate Management Aptitude Test (GMAT) score. A sample of 20 students who had completed 2 years in the program is selected; the results are as follows:

Observation GMAT Score GPI Observation GMAT Score GPI

1 688 3.72 11 567 3.07

2 647 3.44 12 542 2.86

3 652 3.21 13 551 2.91

4 608 3.29 14 573 2.79

5 680 3.91 15 536 3.00

6 617 3.28 16 639 3.55

7 557 3.02 17 619 3.47

8 599 3.13 18 694 3.60

9 616 3.45 19 718 3.88

10 594 3.33 20 759 3.76

(Hint: First determine which are the independent and dependent variables.)

a. Plot a scatter diagram [Using Excel] and, assuming a linear relationship, use the least-squares method to find the regression coefficients b0 & b1.

The question says that we would like to be able to predict the GPI based on the GMAT score. This means that the GPI is the dependent (y) variable, and the GMAT score is the independent variable (x).

I put these variables into an Excel spreadsheet, and made a scatterplot:

Then, I went to chart  add trendline  linear, making sure that I checked the box specifying that the equation show up on the screen. The regression equation for these data is:

y = 0.0049x + 0.3003

The coefficients are:

b0 = 0.3003 (y-intercept)

b1 = 0.0049 (slope)

b. Interpret the meaning of the Y intercept b0 and the slope b1 in this problem.

The ...

#### Solution Summary

The solution includes answers to all parts of the question, which together form a complete linear regression analysis. Step-by-step explanations are included.

Calculations for Simple Linear Regression

6. Following is a table with the set of x-values and their corresponding y-values filled in

xi yi xi2 xiyi

1 6

1 7

2 5

3 5

4 4

6 2

7 2

totals __ __ ___ ____

(sum of all values in each column)

a. Complete the table.

b. Find SSxy

c. Find SSxx

d. Find the least squares line.

e. Calculate SSE

f. Calculate s^2

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