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Regression Models

To accompany Quantitative Analysis for Management, Twelfth Edition,

by Render, Stair, Hanna and Hale

Power Point slides created by Jeff Heyl

After completing this chapter, students will be able to:

LEARNING OBJECTIVES

4 – 2

Identify variables and use them in a regression model

Develop simple linear regression equations from sample data and interpret the slope and intercept

Compute the coefficient of determination and the coefficient of correlation and interpret their meanings

Interpret the F test in a linear regression model

List the assumptions used in regression and use residual plots to identify problems

After completing this chapter, students will be able to:

LEARNING OBJECTIVES

4 – 3

Develop a multiple regression model and use it for prediction purposes

Use dummy variables to model categorical data

Determine which variables should be included in a multiple regression model

Transform a nonlinear function into a linear one for use in regression

Understand and avoid common mistakes made in the use of regression analysis

4 – 4

4.1 Introduction

4.2 Scatter Diagrams

4.3 Simple Linear Regression

4.4 Measuring the Fit of the Regression Model

4.5 Assumptions of the Regression Model

4.6 Testing the Model for Significance

CHAPTER OUTLINE

4 – 5

4.7 Using Computer Software for Regression

4.8 Multiple Regression Analysis

4.9 Binary or Dummy Variables

4.10 Model Building

4.11 Nonlinear Regression

4.12 Cautions and Pitfalls in Regression Analysis

CHAPTER OUTLINE

Introduction

Regression analysis – very valuable tool for a manager

Understand the relationship between variables

Predict the value of one variable based on another variable

Simple linear regression models have only two variables

Multiple regression models have more than one independent variable

4 – 6

Introduction

Variable to be predicted is called the dependent variable or response variable

Value depends on the value of the independent variable(s)

Explanatory or predictor variable

4 – 7

Independent variable

Dependent variable

Independent variable

= +

Scatter Diagram

Scatter diagram or scatter plot often used to investigate the relationship between variables

Independent variable normally plotted on X axis

Dependent variable normally plotted on Y axis

4 – 8

Triple A Construction

Triple A Construction renovates old homes

The dollar volume of renovation work is dependent on the area payroll

4 – 9

TRIPLE A’S SALES ($100,000s)	LOCAL PAYROLL ($100,000,000s)
6	3
8	4
9	6
5	4
4.5	2
9.5	5

TABLE 4.1

Triple A Construction

Figure 4.1 – Scatter Diagram

4 – 10

| | | | | | | | |

0 1 2 3 4 5 6 7 8

12 –

10 –

8 –

6 –

4 –

2 –

0 –

Sales ($100,000)

Payroll ($100 million)

Simple Linear Regression

Regression models used to test relationships between variables

Random error

4 – 11

where

Y = dependent variable (response)

X = independent variable (predictor or explanatory)

0 = intercept (value of Y when X = 0)

1 = slope of the regression line

e = random error

Simple Linear Regression

True values for the slope and intercept are not known

Estimated using sample data

4 – 12

where

Y = predicted value of Y

b0 = estimate of β0, based on sample results

b1 = estimate of β1, based on sample results

Triple A Construction

Predict sales based on area payroll

Y = Sales

X = Area payroll

The line Figure 4.1 minimizes the errors

4 – 13

Error = (Actual value) – (Predicted value)

Regression analysis minimizes the sum of squared errors

Least-squares regression

Triple A Construction

Formulas for simple linear regression, intercept and slope

4 – 14

Triple A Construction

4 – 15

Y	X	(X – X)2	(X – X)(Y – Y)
6	3	(3 – 4)2 = 1	(3 – 4)(6 – 7) = 1
8	4	(4 – 4)2 = 0	(4 – 4)(8 – 7) = 0
9	6	(6 – 4)2 = 4	(6 – 4)(9 – 7) = 4
5	4	(4 – 4)2 = 0	(4 – 4)(5 – 7) = 0
4.5	2	(2 – 4)2 = 4	(2 – 4)(4.5 – 7) = 5
9.5	5	(5 – 4)2 = 1	(5 – 4)(9.5 – 7) = 2.5
ΣY = 42 Y = 42/6 = 7	ΣX = 24 X = 24/6 = 4	Σ(X – X)2 = 10	Σ(X – X)(Y – Y) = 12.5

TABLE 4.2 – Regression calculations

Triple A Construction

Regression calculations

4 – 16

Therefore

Regression calculations

Triple A Construction

4 – 17

Therefore

sales = 2 + 1.25(payroll)

If the payroll next year is $600 million

Measuring the Fit of the Regression Model

Regression models can be developed for any variables X and Y

How helpful is the model in predicting Y?

With average error positive and negative errors cancel each other out

Three measures of variability

SST – Total variability about the mean

SSE – Variability about the regression line

SSR – Total variability that is explained by the model

4 – 18

Measuring the Fit of the Regression Model

Sum of squares total

Sum of squares error

Sum of squares regression

An important relationship

4 – 19

Measuring the Fit of the Regression Model

4 – 20

Y	X	(Y – Y)2	Y	(Y – Y)2	(Y – Y)2
6	3	(6 – 7)2 = 1	2 + 1.25(3) = 5.75	0.0625	1.563
8	4	(8 – 7)2 = 1	2 + 1.25(4) = 7.00	1	0
9	6	(9 – 7)2 = 4	2 + 1.25(6) = 9.50	0.25	6.25
5	4	(5 – 7)2 = 4	2 + 1.25(4) = 7.00	4	0
4.5	2	(4.5 – 7)2 = 6.25	2 + 1.25(2) = 4.50	0	6.25
9.5	5	(9.5 – 7)2 = 6.25	2 + 1.25(5) = 8.25	1.5625	1.563
		∑(Y – Y)2 = 22.5	∑(Y – Y)2	= 6.875	∑(Y – Y)2 = 15.625
Y = 7		SST = 22.5	SSE	= 6.875	SSR = 15.625

TABLE 4.3 – Sum of Squares for Triple A Construction

Sum of squares total

Sum of squares error

Sum of squares regression

An important relationship

Measuring the Fit of the Regression Model

4 – 21

For Triple A Construction

SST = 22.5

SSE = 6.875

SSR = 15.625

Measuring the Fit of the Regression Model

4 – 22

FIGURE 4.2 – Deviations from the Regression Line and from the Mean

| | | | | | | | |

0 1 2 3 4 5 6 7 8

12 –

10 –

8 –

6 –

4 –

2 –

0 –

Sales ($100,000)

Payroll ($100 million)

Coefficient of Determination

The proportion of the variability in Y explained by the regression equation

The coefficient of determination is r2.

4 – 23

For Triple A Construction

The proportion of the variability in Y explained by the regression equation

The coefficient of determination is r2.

Coefficient of Determination

4 – 24

For Triple A Construction

About 69% of the variability in Y is explained by the equation based on payroll (X)

Correlation Coefficient

An expression of the strength of the linear relationship

Always between +1 and –1

The correlation coefficient is r

4 – 25

For Triple A Construction

Four Values of the Correlation Coefficient

(a) Perfect Positive Correlation: r = +1

(d) Perfect Negative Correlation: r = –1

(b) Positive Correlation: 0 < r < 1

FIGURE 4.3

4 – 26

Assumptions of the Regression Model

With certain assumptions about the errors, statistical tests can be performed to determine the model’s usefulness

Errors are independent

Errors are normally distributed

Errors have a mean of zero

Errors have a constant variance

A plot of the residuals (errors) often highlights glaring violations of assumptions

4 – 27

Residual Plots

FIGURE 4.4A – Pattern of Errors Indicating Randomness

Error

4 – 28

Residual Plots

FIGURE 4.4B – Nonconstant error variance

Error

4 – 29

Residual Plots

FIGURE 4.4C – Errors Indicate Relationship is not Linear

Error

4 – 30

Estimating the Variance

Errors are assumed to have a constant variance ( 2), usually unknown

Estimated using the mean squared error (MSE), s2

4 – 31

where

n = number of observations in the sample

k = number of independent variables

Estimating the Variance

For Triple A Construction

4 – 32

Estimate the standard deviation, s

The standard error of the estimate or the standard deviation of the regression

Testing the Model for Significance

When the sample size is too small, you can get good values for MSE and r2 even if there is no relationship between the variables

Testing the model for significance helps determine if the values are meaningful

Performing a statistical hypothesis test

4 – 33

Testing the Model for Significance

We start with the general linear model

4 – 34

If 1 = 0, the null hypothesis is that there is no relationship between X and Y

The alternate hypothesis is that there is a linear relationship (1 ≠ 0)

If the null hypothesis can be rejected, we have proven there is a relationship

We use the F statistic

Testing the Model for Significance

The F statistic is based on the MSE and MSR

4 – 35

where

k = number of independent variables in the model

Describes an F distribution with:

degrees of freedom for the numerator = df1 = k

degrees of freedom for the denominator = df2 = n – k – 1

The F statistic is

Testing the Model for Significance

If there is very little error, MSE would be small and the F statistic would be large – model is useful

If the F statistic is large, the significance level (p-value) will be low, – unlikely would have occurred by chance

When the F value is large, we can reject the null hypothesis and accept that there is a linear relationship between X and Y and the values of the MSE and r2 are meaningful

4 – 36

Steps in a Hypothesis Test

Specify null and alternative hypotheses

4 – 37

Select the level of significance ()

Common values are 0.01 and 0.05.

Calculate the value of the test statistic

Steps in a Hypothesis Test

Make a decision using one of the following methods

4 – 38

Reject the null hypothesis if the test statistic is greater than the F value from the table in Appendix D. Otherwise, do not reject the null hypothesis:

Reject the null hypothesis if the observed significance level, or p-value, is less than the level of significance (). Otherwise, do not reject the null hypothesis:

Triple A Construction

Step 1

4 – 39

H0: 1 = 0 (no linear relationship between X and Y)

H1: 1 ≠ 0 (linear relationship exists between X and Y)

Step 2

Select  = 0.05

Step 3

– Calculate the value of the test statistic

Triple A Construction

Step 4

Reject the null hypothesis if the test statistic is greater than the F value in Appendix D

4 – 40

df1 = k = 1

df2 = n – k – 1 = 6 – 1 – 1 = 4

The value of F associated with a 5% level of significance and with degrees of freedom 1 and 4 is found in Appendix D.

F0.05,1,4 = 7.71

Fcalculated = 9.09

Reject H0 because 9.09 > 7.71

F = 7.71

0.05

9.09

Triple A Construction

We can conclude there is a statistically significant relationship between X and Y

The r2 value of 0.69 means about 69% of the variability in sales (Y) is explained by local payroll (X)

4 – 41

FIGURE 4.5

Analysis of Variance (ANOVA) Table

With software models, an ANOVA table is typically created that shows the observed significance level (p-value) for the calculated F value

This can be compared to the level of significance () to make a decision

4 – 42

	DF	SS	MS	F	SIGNIFICANCE
Regression	k	SSR	MSR = SSR/k	MSR/MSE	P(F > MSR/MSE)
Residual	n - k - 1	SSE	MSE = SSE/(n - k - 1)
Total	n - 1	SST

TABLE 4.4

ANOVA for Triple A Construction

4 – 43

PROGRAM 4.1C – Excel Output for Triple A Construction

P(F > 9.0909) = 0.0394

Using Software

4 – 44

PROGRAM 4.1A – Accessing the Regression Option in Excel

Using Software

4 – 45

PROGRAM 4.1B – Data Input

Using Software

4 – 46

PROGRAM 4.1C – Excel Output

Using Software

4 – 47

PROGRAM 4.2A – Using Excel QM

Using Software

4 – 48

PROGRAM 4.2B – Initializing the Spreadsheet

Using Software

4 – 49

PROGRAM 4.2C – Input and Results

Using Software

4 – 50

PROGRAM 4.3A – QM for Windows Regression Option in Forecasting Module

Using Software

4 – 51

PROGRAM 4.3B – QM for Windows Screen to Initialize the Problem

Using Software

4 – 52

PROGRAM 4.3C – Data Input

Using Software

4 – 53

PROGRAM 4.3D – QM for Windows Output

Multiple Regression Analysis

Extensions to the simple linear model

Models with more than one independent variable

4 – 54

Y = 0 + 1X1 + 2X2 + … + kXk + 

where

Y = dependent variable (response variable)

Xi = ith independent variable (predictor or explanatory variable)

0 = intercept (value of Y when all Xi = 0)

i = coefficient of the ith independent variable

k = number of independent variables

 = random error

Multiple Regression Analysis

To estimate these values, a sample is taken the following equation developed

4 – 55

Where

= predicted value of Y

b0 = sample intercept (an estimate of 0)

bi = sample coefficient of the ith variable (an estimate of i)

Jenny Wilson Realty

Develop a model to determine the suggested listing price for houses based on the size and age of the house

4 – 56

where

= predicted value of dependent variable (selling price)

b0 = Y intercept

X1 and X2 = value of the two independent variables (square footage and age) respectively

b1 and b2 = slopes for X1 and X2 respectively

– Selects a sample of houses that have sold recently and records the data

Jenny Wilson Real Estate Data

4 – 57

SELLING PRICE ($)	SQUARE FOOTAGE	AGE	CONDITION
95,000	1,926	30	Good
119,000	2,069	40	Excellent
124,800	1,720	30	Excellent
135,000	1,396	15	Good
142,000	1,706	32	Mint
145,000	1,847	38	Mint
159,000	1,950	27	Mint
165,000	2,323	30	Excellent
182,000	2,285	26	Mint
183,000	3,752	35	Good
200,000	2,300	18	Good
211,000	2,525	17	Good
215,000	3,800	40	Excellent
219,000	1,740	12	Mint

TABLE 4.5

Jenny Wilson Realty

PROGRAM 4.2A – Input Screen for the Jenny Wilson Realty Multiple Regression Example

4 – 58

Jenny Wilson Realty

Program 4.2B – Excel Output

4 – 59

Evaluating Multiple Regression Models

Similar to simple linear regression models

The p-value for the F test and r2 interpreted the same

The hypothesis is different because there is more than one independent variable

The F test is investigating whether all the coefficients are equal to 0 at the same time

4 – 60

Evaluating Multiple Regression Models

To determine which independent variables are significant, tests are performed for each variable

The test statistic is calculated and if the p-value is lower than the level of significance (), the null hypothesis is rejected

4 – 61

Jenny Wilson Realty

Full model is statistically significant

Useful in predicting selling price

4 – 62

Are both variables significant?

For X1 (square footage)

p-value for F test = 0.002 r2 = 0.6719

For a = 0.05, p-value = 0.0013 null hypothesis is rejected

For X1 (age)

For a = 0.05, p-value = 0.0039 null hypothesis is rejected

Full model is statistically significant

Useful in predicting selling price

Jenny Wilson Realty

4 – 63

Are both variables significant?

For X1 – square footage

p-value for F test = 0.002 r2 = 0.6719

For a = 0.05, p-value = 0.0013 null hypothesis is rejected

For X1 – age

For a = 0.05, p-value = 0.0039 null hypothesis is rejected

Both square footage and age are helpful in predicting the selling price

Binary or Dummy Variables

Binary (or dummy or indicator) variables are special variables created for qualitative data

A dummy variable is assigned a value of 1 if a particular condition is met and a value of 0 otherwise

The number of dummy variables must equal one less than the number of categories of the qualitative variable

4 – 64

Jenny Wilson Realty

A better model can be developed if information about the condition of the property is included

4 – 65

X3 = 1 if house is in excellent condition

= 0 otherwise

X4 = 1 if house is in mint condition

= 0 otherwise

Two dummy variables are used to describe the three categories of condition

No variable is needed for “good” condition since if both X3 and X4 = 0, the house must be in good condition

Jenny Wilson Realty

PROGRAM 4.5A – Excel Input Screen with Dummy Variables

4 – 66

Jenny Wilson Realty

PROGRAM 4.5B – Excel Output with Dummy Variables

4 – 67

Jenny Wilson Realty

PROGRAM 4.5B – Excel Output with Dummy Variables

4 – 68

Coefficient of determination, r2 = 0.898

Model Building

The best model is a statistically significant model with a high r2 and few variables

As more variables are added to the model, the r2 value increases

For this reason, the adjusted r2 value is often used to determine the usefulness of an additional variable

The adjusted r2 takes into account the number of independent variables in the model

4 – 69

Model Building

The formula for r2

The formula for adjusted r2

As the number of variables increases, the adjusted r2 gets smaller unless the increase due to the new variable is large enough to offset the change in k

4 – 70

The formula for r2

The formula for adjusted r2

As the number of variables increases, the adjusted r2 gets smaller unless the increase due to the new variable is large enough to offset the change in k

Model Building

4 – 71

In general, if a new variable increases the adjusted r2, it should probably be included in the model

Model Building

Stepwise regression systematically adds or deletes independent variables

A forward stepwise procedure puts the most significant variable in first, adds the next variable that will improve the model the most

Backward stepwise regression begins with all the independent variables and deletes the least helpful

4 – 72

Model Building

In some cases variables contain duplicate information

When two independent variables are correlated, they are said to be collinear

When more than two independent variables are correlated, multicollinearity exists

When multicollinearity is present, hypothesis tests for the individual coefficients are not valid but the model may still be useful

4 – 73

Nonlinear Regression

In some situations, variables are not linear

Transformations may be used to turn a nonlinear model into a linear model

Linear relationship

Nonlinear relationship

4 – 74

Colonel Motors

Use regression analysis to improve fuel efficiency

Study the impact of weight on miles per gallon (MPG)

4 – 75

MPG	WEIGHT (1,000 LBS.)	MPG	WEIGHT (1,000 LBS.)
12	4.58	20	3.18
13	4.66	23	2.68
15	4.02	24	2.65
18	2.53	33	1.70
19	3.09	36	1.95
19	3.11	42	1.92

TABLE 4.6

Colonel Motors

4 – 76

FIGURE 4.6A – Linear Model for MPG Data

| | | | | |

1.00 2.00 3.00 4.00 5.00

45 –

40 –

35 –

30 –

25 –

20 –

15 –

10 –

5 –

0 –

Weight (1,000 lb.)

MPG

Colonel Motors

PROGRAM 4.6 – Excel Output for Linear Regression Model with MPG Data

4 – 77

Useful model

Small F test for significance

Good r2 value

Colonel Motors

4 – 78

FIGURE 4.6B – Nonlinear Model for MPG Data

| | | | | |

1.00 2.00 3.00 4.00 5.00

45 –

40 –

35 –

30 –

25 –

20 –

15 –

10 –

5 –

0 –

Weight (1,000 lb.)

MPG

Colonel Motors

The nonlinear model is a quadratic model

The easiest approach – develop a new variable

4 – 79

New model

Colonel Motors

PROGRAM 4.7 – Excel Output for Nonlinear Regression Model with MPG Data

4 – 80

Improved model

Small F test for significance

Adjusted r2 and r2 both increased

Cautions and Pitfalls

If the assumptions are not met, the statistical test may not be valid

Correlation does not necessarily mean causation

Multicollinearity makes interpreting coefficients problematic, but the model may still be good

Using a regression model beyond the range of X is questionable, as the relationship may not hold outside the sample data

4 – 81

Cautions and Pitfalls

A t-test for the intercept (b0) may be ignored as this point is often outside the range of the model

A linear relationship may not be the best relationship, even if the F test returns an acceptable value

A nonlinear relationship can exist even if a linear relationship does not

Even though a relationship is statistically significant it may not have any practical value

4 – 82

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.

1.6 The Role of Computers and Spreadsheet Models in the Quantitative Analysis Approach

1.7 Possible Problems in the Quantitative Analysis Approach

1.8 Implementation—Not Just the Final Step

1.1 Introduction 1.2 What Is Quantitative Analysis? 1.3 Business Analytics 1.4 The Quantitative Analysis Approach 1.5 How to Develop a Quantitative Analysis Model

CHAPTER OUTLINE

5. Use computers and spreadsheet models to perform quantitative analysis.

6. Discuss possible problems in using quantitative analysis.

7. Perform a break-even analysis.

1. Describe the quantitative analysis approach. 2. Understand the application of quantitative analysis

in a real situation. 3. Describe the three categories of business analytics. 4. Describe the use of modeling in quantitative

analysis.

After completing this chapter, students will be able to:

Introduction to Quantitative Analysis

1CHAPTER

LEARNING OBJECTIVES

M01_REND7331_12_SE_C01_pp2.indd 1 01/10/13 9:50 AM

1.6 The Role of Computers and Spreadsheet Models in the Quantitative Analysis Approach

1.7 Possible Problems in the Quantitative Analysis Approach

1.8 Implementation—Not Just the Final Step

1.1 Introduction 1.2 What Is Quantitative Analysis? 1.3 Business Analytics 1.4 The Quantitative Analysis Approach 1.5 How to Develop a Quantitative Analysis Model

CHAPTER OUTLINE

5. Use computers and spreadsheet models to perform quantitative analysis.

6. Discuss possible problems in using quantitative analysis.

7. Perform a break-even analysis.

1. Describe the quantitative analysis approach. 2. Understand the application of quantitative analysis

in a real situation. 3. Describe the three categories of business analytics. 4. Describe the use of modeling in quantitative

analysis.

After completing this chapter, students will be able to:

Introduction to Quantitative Analysis

1CHAPTER

LEARNING OBJECTIVES

M01_REND7331_12_SE_C01_pp2.indd 1 01/10/13 9:50 AM

Y = β0 +β1X +ε

Y=b

X+e

Ŷ = b0 +b1X

Y=b

e =Y −Ŷ

e=Y-

Ŷ = b0 +b1X

Y=b

X = X∑ n

= average (mean) of X values

Y = Y∑ n

= average (mean) of Y values

b1 = (X − X∑ )(Y −Y) (X − X)2∑

b0 =Y −b1X

=average (mean) of X values

=average (mean) of Y values

(X-X

)(Y-Y)

(X-X)

=Y-b

X = X∑ 6

= 24 6 = 4

Y = Y∑ 6

= 42 6 =7

b1 = (X – X∑ )(Y –Y) (X – X)2∑

= 12.5 10

=1.25

b0 =Y −b1X = 7 – (1.25)(4) = 2

(X–X

)(Y–Y)

(X–X)

12.5

=1.25

=Y-b

X=7 – (1.25)(4)=2

Ŷ = 2 + 1.25X

Y=2 + 1.25X

Ŷ = 2 + 1.25(6) = 9.5 or $ 950,000

Y=2 + 1.25(6)=9.5 or $ 950,000

SST = (Y −Y∑ )2

SST=(Y-Y

)

SSE= e2 = (Y −Ŷ)2∑∑

SSE=e

=(Y-

åå

SSR= (Ŷ −Y)2∑

SSR=(

Y-Y)

SST = SSR + SSE

SST=SSR + SSE

Ŷ = 2 + 1.25X

Y = 2 + 1.25X

Y – Ŷ

Y –

Ŷ – Y

Y – Y

r2 = SSR SST

=1 – SSE SST

SSR

SST

=1 –

SSE

SST

r2 = 15.625 22.5

=0.6944

15.625

22.5

=0.6944

r =± r2

r=±r

r = 0.6944 =0.8333

r=0.6944=0.8333

s2 =MSE= SSE n−k −1

=MSE=

SSE

n-k-1

s2 =MSE= SSE n−k −1

= 6.8750 6-1-1

= 6.8750 4

=1.7188

=MSE=

SSE

n-k-1

6.8750

6-1-1

6.8750

=1.7188

s = MSE = 1.7188 = 1.31

s = MSE = 1.7188= 1.31

Y = β0 +β1X +ε

Y=b

X+e

MSR= SSR k

MSR=

SSR

F = MSR MSE

MSR

MSE

H0 :β1 =0 H1 :β1 ≠0

¹0

F = MSR MSE

MSR

MSE

Reject if Fcalculated > Fα,df1,df2 df1 = k df2 = n−k −1

Reject if F

calculated

a,df

,df

=n-k-1

p-value= P(F > calculated test statistic) Reject if p-value<α

p-value=P(F>calculated test statistic)

Reject if p-value<a

MSR= SSR k

= 15.6250

1 =15.6250

F = MSR MSE

= 15.6250 1.7188

=9.09

MSR=

SSR

15.6250

=15.6250

MSR

MSE

15.6250

1.7188

=9.09

Enter the past observations of Y and X. Results appear automatically.

To forecast Y based on any value of X, simply input the value of X here.

Correlation coef!cient is given here.

The intercept and slope are shown here.

After selecting the Forecasting module, click New and select Least Squares – Simple and Multiple Regression.

Specify the number of observations.

Specify the number of variables.

Click OK.

Ŷ = b0 +b1X1 +b2X2 +...+bkXk

Y=b

+...+b

Ŷ

Ŷ = b0 +b1X1 +b2X2

Y=b

Ŷ

The columns containing data for the X variables should be adjoining.

The X range is from the start of the !rst column to the bottom of the second column.

Ŷ = b0 +b1X1 +b2X2 =146,630.89+43.82X1 −2898.69X2

Y=b

=146,630.89+43.82X

-2898.69X

The coef!cients are here.

The coef!cient of determination is 0.67.

A low Signi!cance F value indicates that the model is statistically signi!cant.

Ŷ =121,658 + 56.43X1 – 3,962X2 + 33,162X3 + 47,369X4

Y=121,658 + 56.43X

– 3,962X

+ 33,162X

+ 47,369X

r2 = SSR SST

=1− SSE SST

SSR

SST

=1-

SSE

SST

Adjusted r2 =1− SSE / (n−k −1)

SST / (n−1)

Adjusted r

=1-

SSE/(n-k-1)

SST/(n-1)

Ŷ = 47.6−8.2X1 or MPG = 47.6 – 8.2 (weight in 1,000 lb.)

Y=47.6-8.2X

or MPG= 47.6 – 8.2 (weight in 1,000 lb.)

X2 = (weight) 2

=(weight)

Ŷ = b0 +b1X1 +b2X2

Y=b

Ŷ =79.8−30.2X1+3.4X2

Y=79.8-30.2X

+3.4X