Statistics for Managers Using Microsoft Excel/SPSS · 13/02/2016 · Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression ... Statistics for

© 1999 Prentice-Hall, Inc. Chap. 13 - 1

Statistics for Managers

Using Microsoft Excel/SPSS

Chapter 13

The Simple Linear Regression

Model and Correlation


Chapter Topics

• Types of Regression Models

• Determining the Simple Linear Regression Equation

• Measures of Variation in Regression and Correlation

• Assumptions of Regression and Correlation

• Residual Analysis and the Durbin-Watson Statistic

• Estimation of Predicted Values

• Correlation - Measuring the Strength of the Association


Purpose of Regression and Correlation Analysis

• Regression Analysis is Used Primarily for

Prediction

A statistical model used to predict the values of a

dependent or response variable based on values of

at least one independent or explanatory variable

Correlation Analysis is Used to Measure

Strength of the Association Between

Numerical Variables


The Scatter Diagram

0

20

40

60

0 20 40 60

X

Y

Plot of all (Xi , Yi) pairs


Types of Regression Models

Positive Linear Relationship

Negative Linear Relationship

Relationship NOT Linear

No Relationship


Simple Linear Regression Model

iii XY 10

Y intercept

Slope

• The Straight Line that Best Fit the Data

• Relationship Between Variables Is a Linear Function

Random

Error

Dependent

(Response)

Variable

Independent

(Explanatory)

Variable


i = Random Error

Y

X

Population

Linear Regression Model

Observed

Value

Observed Value

m YX i X 0 1

Y X i i i 0 1


Sample Linear Regression Model

ii XbbY 10

Yi

= Predicted Value of Y for observation i

Xi = Value of X for observation i

b0 = Sample Y - intercept used as estimate of

the population 0

b1 = Sample Slope used as estimate of the

population 1


Simple Linear Regression Equation: Example

You wish to examine the

relationship between the

square footage of produce

stores and its annual sales.

Sample data for 7 stores

were obtained. Find the

equation of the straight

line that fits the data best

Annual Store Square Sales Feet ($000)

1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760


Scatter Diagram Example

0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 0 0 0 0

1 2 0 0 0

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0

S q u a re F e e t

An

nu

al

Sa

les (

$0

00

)

Excel Output


Equation for the Best Straight Line

i

ii

X..

XbbY

48714151636

10

From Excel Printout:

C o effic ien ts

I n te r c e p t 1 6 3 6 . 4 1 4 7 2 6

X V a r i a b l e 1 1 . 4 8 6 6 3 3 6 5 7


Graph of the Best Straight Line

0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 0 0 0 0

1 2 0 0 0

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0

S q u a re F e e t

An

nu

al

Sa

les (

$0

00

)


Interpreting the Results

Yi = 1636.415 +1.487Xi

The slope of 1.487 means for each increase of one

unit in X, the Y is estimated to increase 1.487units.

For each increase of 1 square foot in the size of the

store, the model predicts that the expected annual

sales are estimated to increase by $1487.


Measures of Variation: The Sum of Squares

SST = Total Sum of Squares

•measures the variation of the Yi values around their

mean Y

SSR = Regression Sum of Squares

•explained variation attributable to the relationship

between X and Y

SSE = Error Sum of Squares

•variation attributable to factors other than the

relationship between X and Y

_


Measures of Variation: The Sum of Squares

Xi

Y

X

Y

SST = (Yi - Y)2

SSE =(Yi - Yi )2

SSR = (Yi - Y)2

_

_

_


d f S S

R e g r e ssi o n 1 3 0 3 8 0 4 5 6 . 1 2

R e si d u a l 5 1 8 7 1 1 9 9 . 5 9 5

T o ta l 6 3 2 2 5 1 6 5 5 . 7 1

Measures of Variation The Sum of Squares:

Example

Excel Output for Produce Stores

SSR SSE SST


The Coefficient of Determination

SSR regression sum of squares

SST total sum of squares r2 = =

Measures the proportion of variation that is

explained by the independent variable X in

the regression model


Coefficients of Determination

(r2) and Correlation (r)

r2 = 1, r2 = 1,

r2 = .8, r2 = 0, Y

Y i = b 0 + b 1 X i

X

^

Y

Y i = b 0 + b 1 X i

X

^ Y

Y i = b 0 + b 1 X i X

^

Y

Y i = b 0 + b 1 X i

X

^

r = +1 r = -1

r = +0.9 r = 0


Standard Error of Estimate

2

n

SSESyx

2

1

2

n

)YY(n

iii

=

The standard deviation of the variation of

observations around the regression line


R e g re ssio n S ta tistic s

M u lt ip le R 0 . 9 7 0 5 5 7 2

R S q u a re 0 . 9 4 1 9 8 1 2 9

A d ju s t e d R S q u a re 0 . 9 3 0 3 7 7 5 4

S t a n d a rd E rro r 6 1 1 . 7 5 1 5 1 7

O b s e rva t io n s 7

Measures of Variation:

Example

Excel Output for Produce Stores

r2 = .94 Syx 94% of the variation in annual sales can be

explained by the variability in the size of the

store as measured by square footage


Linear Regression

Assumptions

1. Normality

Y Values Are Normally Distributed For Each

X

Probability Distribution of Error is Normal

2. Homoscedasticity (Constant Variance)

3. Independence of Errors

For Linear Models


Variation of Errors Around the Regression Line

X1

X2

X

Y

f(e) y values are normally distributed

around the regression line.

For each x value, the “spread” or

variance around the regression

line is the same.

Regression Line


Residual Analysis

• Purposes

Examine Linearity

Evaluate violations of assumptions

• Graphical Analysis of Residuals

Plot residuals Vs. Xi values

Difference between actual Yi & predicted Yi

Studentized residuals:

Allows consideration for the magnitude of the

residuals


Residual Analysis for Linearity

Not Linear Linear

X

e e

X


Residual Analysis for Homoscedasticity

Heteroscedasticity Homoscedasticity

Using Standardized Residuals

SR

X

SR

X


R e s id u a l P lo t

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0

S q u a r e F e e t

Residual Analysis:

Computer Output Example

Produce Stores

Excel Output

Observation Predicted Y Residuals

1 4202.344417 -521.3444173

2 3928.803824 -533.8038245

3 5822.775103 830.2248971

4 9894.664688 -351.6646882

5 3557.14541 -239.1454103

6 4918.90184 644.0981603

7 3588.364717 171.6352829


The Durbin-Watson Statistic

•Used when data is collected over time to detect

autocorrelation (Residuals in one time period

are related to residuals in another period)

•Measures Violation of independence assumption

n

ii

n

iii

e

)ee(D

1

2

2

21 Should be close to 2.

If not, examine the model

for autocorrelation.


Residual Analysis for

Independence

Not Independent Independent

X

SR

X

SR


Inferences about the Slope: t Test

• t Test for a Population Slope

Is a Linear Relationship Between X & Y ?

1

11

bS

bt

•Test Statistic:

n

ii

YXb

)XX(

SS

1

21

and df = n - 2

•Null and Alternative Hypotheses

H0: 1 = 0 (No Linear Relationship)

H1: 1 0 (Linear Relationship)

Where


Example: Produce Stores

Data for 7 Stores: Regression

Model Obtained:

The slope of this model

is 1.487.

Is there a linear

relationship between the

square footage of a store

and its annual sales?


1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760

Yi = 1636.415 +1.487Xi


t S tat P-value

In te rce p t 3 .6244333 0 .0151488

X V a ria b le 1 9 .009944 0 .0002812

H0: 1 = 0

H1: 1 0

a .05

df 7 - 2 = 7

Critical Value(s):

Test Statistic:

Decision:

Conclusion:

There is evidence of a

relationship. t 0 2.5706 -2.5706

.025

Reject Reject

.025

From Excel Printout

Reject H0

Inferences about the Slope: t Test Example


Inferences about the Slope: Confidence Interval Example

Confidence Interval Estimate of the Slope

b1 tn-2 1bS

Excel Printout for Produce Stores

At 95% level of Confidence The confidence Interval for the

slope is (1.062, 1.911). Does not include 0.

Conclusion: There is a significant linear relationship

between annual sales and the size of the store.

Low er 95% Upper 95%

In te rc e p t 4 7 5 .8 1 0 9 2 6 2 7 9 7 .0 1 8 5 3

X V a r ia b le 11 .0 6 2 4 9 0 3 7 1 .9 1 0 7 7 6 9 4


Estimation of Predicted Values

Confidence Interval Estimate for mXY

The Mean of Y given a particular Xi

n

ii

iyxni

)XX(

)XX(

nStY

1

2

2

2

1

t value from table

with df=n-2

Standard error

of the estimate

Size of interval vary according to

distance away from mean, X.


Estimation of Predicted Values

Confidence Interval Estimate for

Individual Response Yi at a Particular Xi

n

ii

iyxni

)XX(

)XX(

nStY

1

2

2

2

11

Addition of this 1 increased width of

interval from that for the mean Y


Interval Estimates for

Different Values of X

X

Y

X

Confidence Interval

for a individual Yi

A Given X

Confidence

Interval for the

mean of Y

_


Example: Produce Stores

Yi = 1636.415 +1.487Xi

Data for 7 Stores:

Regression Model Obtained:

Predict the annual

sales for a store with

2000 square feet.


1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760


Estimation of Predicted Values: Example

Confidence Interval Estimate for Individual Y

Find the 95% confidence interval for the average annual sales

for stores of 2,000 square feet

n

ii

iyxni

)XX(

)XX(

nStY

1

2

2

2

1

Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)

X = 2350.29 SYX = 611.75 tn-2 = t5 = 2.5706

= 4610.45 980.97

Confidence interval for mean Y


Estimation of Predicted Values: Example

Confidence Interval Estimate for mXY

Find the 95% confidence interval for annual sales of one

particular stores of 2,000 square feet

Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)

X = 2350.29 SYX = 611.75 tn-2 = t5 = 2.5706

= 4610.45 1853.45

Confidence interval for

individual Y

n

ii

iyxni

)XX(

)XX(

nStY

1

2

2

2

11


Correlation: Measuring the

Strength of Association

• Answer ‘How Strong Is the Linear

Relationship Between 2 Variables?’

• Coefficient of Correlation Used

Population correlation coefficient denoted

r (‘Rho’)

Values range from -1 to +1

Measures degree of association

• Is the Square Root of the Coefficient of

Determination


Test of

Coefficient of Correlation

• Tests If There Is a Linear Relationship

Between 2 Numerical Variables

• Same Conclusion as Testing Population

Slope 1

• Hypotheses

H0: r = 0 (No Correlation)

H1: r 0 (Correlation)


Chapter Summary

• Described Types of Regression Models

• Determined the Simple Linear Regression Equation

• Provided Measures of Variation in Regression and Correlation

• Stated Assumptions of Regression and Correlation

• Described Residual Analysis and the Durbin-Watson Statistic

• Provided Estimation of Predicted Values

• Discussed Correlation - Measuring the Strength of the Association

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Statistics for Managers Using Microsoft Excel/SPSS · 13/02/2016 · Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression ... Statistics for