A1 Business and Technical College Module Multiple Regression Excel Sheet download the files down below andfollow all the steps pleasethank you>>>>>>>>>>>>a The CollegeGPA worksheet contains information about 224 students who entered a

university in a particular year and were planning to major in computer science. The data

were collected to see if information available before students enroll can predict success

in the early university years. The response variable (Y) is a student’s cumulative grade

point average (GPA) after three semesters. Among the predictors are average high

school grades in mathematics (HSM), science (HSS), and English (HSE). We expect

high school grades in these three subjects will help predict College GPA.

Download and go over the attached pdf file for information about multiple regression.

1.

Use Excel to obtain the regression estimates and write the regression equation.

2.

Interpret each regression coefficient.

3.

Make a prediction for a straight-C student’s cumulative grade point average (GPA) when HSM = 4, HSS = 4,

and HSE = 4.

4.

What is the coefficient of determination (R2). Based on the R2, how would you describe the fit of the model?

5.

What is the adjusted R2? Does the model contain useless predictors?

6.

What is the F statistic and its p-value? Based on the p-value, is the regression significant overall at the 0.05

significance level?

7.

Fill the first two columns of the table in the Results worksheet of the CollegeGPA with the t statistic and the pvalue for the t statistic.

8.

Based on the p-value for the t statistic, fill the last column of the table to indicate whether these regression

coefficients are significant or not at the 0.05 significance level.

Include all your answers within the workbook. Rename the assignment file as

YourLastName_Assignment9.xlsx

HSM

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HSS

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HSE

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GPA

3.32

2.26

2.35

2.08

3.38

3.29

3.21

2.00

3.18

2.34

3.08

3.34

1.40

1.43

2.48

3.73

3.80

4.00

2.00

3.74

2.32

2.79

3.21

3.08

3.75

3.16

2.73

3.06

1.07

3.35

1.82

3.12

2.25

2.93

3.16

2.83

3.10

3.07

2.87

3.61

2.17

1.95

3.35

3.58

2.75

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3.26

3.41

3.67

3.81

3.30

2.30

3.62

3.20

2.55

2.82

3.25

2.21

2.50

3.03

1.92

4.00

1.81

2.70

2.96

2.76

2.71

3.40

2.65

2.48

3.86

2.62

3.72

3.50

2.83

3.06

4.00

3.70

2.81

1.93

3.70

2.96

2.64

3.09

3.00

2.97

2.81

3.32

3.40

1.84

0.40

2.88

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2.77

2.26

2.03

2.43

2.63

1.66

3.41

2.12

3.33

1.69

2.46

1.59

1.14

0.65

2.12

2.82

2.34

2.11

1.34

2.53

2.75

3.14

2.25

1.00

2.79

2.39

2.15

0.75

3.06

2.50

2.78

2.44

1.11

3.12

2.17

0.12

2.00

3.22

1.88

2.04

2.58

2.16

2.50

1.85

1.46

2.95

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0.80

0.91

2.67

2.51

1.79

2.42

0.58

3.00

2.76

3.35

3.80

2.38

2.58

3.18

2.87

3.16

3.07

3.68

3.34

1.93

2.43

3.28

3.66

2.29

2.19

3.06

3.41

3.14

2.85

3.47

3.44

3.90

3.65

1.32

3.23

2.86

2.51

2.86

3.34

3.33

3.69

1.80

2.57

2.28

1.60

2.00

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1.69

3.06

2.75

2.62

0.39

2.44

3.46

2.37

1.25

2.80

2.14

2.45

2.71

2.59

2.93

2.53

1.95

3.39

2.69

1.94

3.00

2.09

1.85

3.34

2.25

4.00

2.72

2.61

2.32

3.39

3.64

1.80

1.52

3.40

2.86

3.32

2.07

0.85

1.86

2.59

2.28

t statistic

Intercept

HSM

HSS

HSE

p-value

Significant or not?

Multiple Regression

Learning Objectives

?

?

?

?

?

?

?

Limitations of bivariate regression

Understand multiple regression

Determine a regression using Microsoft Excel

Interpret a regression

Prediction using regression

Assess the fit of a regression

Determine if the regression parameters are

significant

Limitations of Bivariate Regression

In a bivariate regression, a low R2 does not

mean that X and Y are not related

? The correct independent variable(s) were not

included

? The model may be too simplistic

? The estimates are thus biased

? Bivariate regression is only used when

? There is a compelling need for a single model

? A single logical predictor stands out as doing

a very good job all by itself

?

Multiple Regression

In an attempt to improve the fit and to specify a

correct model

? Add additional predictors (X1, X2, X3,
.)

? Multiple or multivariate regression

? Extends bivariate regression to include several

independent variables

?

Data Format for a Multiple Regression

To obtain a fitted regression, we need

? n observed values of the response variable Y

and its proposed predictors X1, X2, …, Xk

? A multivariate data set

? Single column of Y-values and k columns of Xvalues

? In Excel

? You are required to have X data in contiguous

columns

?

Data Format for a Multiple Regression

Multiple Regression

?

In the Sales example we went over for the

tutorial

? A personnel researcher wanted to find out

how well sales volume of salespersons could

be predicted from a score derived from

demographic information (Biodata Score) and

a Sales Aptitude test score

? Dependent variable sales

? Independent variables Biodata scores and

Sales Aptitude test score

Determining a Regression

We will use the Excel output to determine the

intercept and the slope

? Refer the results for the Sales data on next slide

? We write the regression equation

?

or

Determining a Regression

Interpreting a Regression

?

Each regression coefficient bj shows

? The change in the expected value of Y for a

unit change in Xj, while holding everything else

constant

Interpreting a Regression

?

For example

? Holding everything else constant

? Each additional square foot adds about $171

to the average selling price

? Each additional square feet of lot size adds

$6,780 to the average selling price

? Each additional baths adds $15,530 to the

average selling price

Interpreting a Regression

?

Looking at the Sales regression line

? Holding everything else constant

? Each unit increase in Biodata score adds

$6490 to the average sales

? Each unit increase in Sales Aptitude adds

$2640 to the average sales

Predicting Using Regression

The fitted model is used to make predictions

for various assumed predictor values

? What would be the expected selling price of a

2,800 square foot home with 2-1/2 baths on a

lot with 18,500 square feet?

SqFt = 2800

LotSize = 18.5

Baths = 2.5

?

Predicting Using Regression

?

Looking at the Sales regression line

?

Lets find out the sales volume of a salesperson

with biodata score of 20 and a sales aptitude

score of 20

Assessing the Fit of a Regression

?

We assess the fit of a regression in two ways

? The coefficient of determination (R2)

? Adjusted R2

? F-statistic

Coefficient of Determination

Denoted by R2

? Lies in the range 0 ? R2 ? 1

? The closer R2 is to 1, the better the fit of the

regression model

? Often expressed as a percent of variation

explained

? Looking at the Sales regression line

? The value for R2 is in cell B5

? R2 = 0.5996 or 59.96%

?

Coefficient of Determination

Coefficient of Determination

?

Interpretation

? This means that 59.96% of the variation in

sales is explained by the two factors

? While this indicates a somewhat good fit

? There is some unexplained variation

? Adding more predictors

? Can never decrease the R2

? Will generally raise the R2

? However, when R2 is high

? There is not a lot of room for improvement

Adjusted R2

In multiple regression, it is possible to increase

R2 by including additional predictors

? This tempts us to believe that we should

include many predictors to get a better fit

? To discourage this tactic, called overfitting the

model

? An adjustment can be made to the R2 statistic

to penalize the inclusion of useless predictors

? Adjusted coefficient of determination

?

Adjusted R2

R2adj is always less than or equal to R2

? As predictors are added

? R2 cannot decline and will rise

? R2adj may rise, remain the same, or fall

? Depending on whether the added

predictors increase R2 enough to offset the

penalty

? If R2adj is substantially smaller than R2

? Suggests that the model contains useless

predictors

?

Adjusted R2

The adjusted R2 can be found in cell B6 of the

Sales data

R2adj = 52.68%

? The R2adj is close to R2

? Suggests that the model does not contain

useless predictors

?

Adjusted R2

F-Statistic for Overall Fit

The F-statistic is a measure of overall fit

? A larger F statistic indicates a better fit

? The p-value is used to determine whether the

regression is significant

? The regression is significant if the p-value ?

0.05

?

F-Statistic for Overall Fit

?

Looking at the Sales regression line

? The F-statistic is in cell E12

? F-statistic = 8.24

? The p-value for the F-statistic is in cell F12

? p-value = 0.0065

? Since the p-value is less than 0.05

? We conclude that the regression is

significant

F-Statistic for Overall Fit

Are the Parameters Significant?

We can also determine whether the regression

parameters are significant

? The regression parameters are significant if the

associated p-value ? 0.05

? We can obtain the t-statistic and p-values for

the slope and intercept from the regression

output from Excel

?

Are the Parameters Significant?

?

Looking at the sales regression line

? The t-statistic and associated p-values for the

regression parameters can be found in the

range D17:E19

Are the Parameters Significant?

Are the Parameters Significant?

Parameter

Intercept

Biodata Score

Sales Aptitude

t-test

p-value

Significant?

– 0.131197677 0.897987708

No

1.565066696 0.145862558

No

0.901971106 0.386396122

No

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