10-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)

10-1

COMPLETE COMPLETE BUSINESS BUSINESS

STATISTICSSTATISTICSbyby

AMIR D. ACZELAMIR D. ACZEL

&&

JAYAVEL SOUNDERPANDIANJAYAVEL SOUNDERPANDIAN

66thth edition (SIE) edition (SIE)

10-2

Chapter 10 Chapter 10

Simple Linear Simple Linear Regression and Regression and

CorrelationCorrelation

10-3

• Using Statistics• The Simple Linear Regression Model• Estimation: The Method of Least Squares• Error Variance and the Standard Errors of Regression

Estimators• Correlation• Hypothesis Tests about the Regression Relationship• How Good is the Regression?• Analysis of Variance Table and an F Test of the

Regression Model• Residual Analysis and Checking for Model Inadequacies• Use of the Regression Model for Prediction• The Solver Method for Regression

Simple Linear Regression and CorrelationSimple Linear Regression and Correlation1010

10-4

• Determine whether a regression experiment would be useful in a given instance

• Formulate a regression model• Compute a regression equation• Compute the covariance and the correlation

coefficient of two random variables• Compute confidence intervals for regression

coefficients• Compute a prediction interval for the dependent

variable

LEARNING OBJECTIVESLEARNING OBJECTIVES1010

After studying this chapter, you should be able to:After studying this chapter, you should be able to:

10-5

• Test hypothesis about a regression coefficients• Conduct an ANOVA experiment using regression

results• Analyze residuals to check if the assumptions about the

regression model are valid• Solve regression problems using spreadsheet templates• Apply covariance concept to linear composites of

random variables• Use LINEST function to carry out a regression

LEARNING OBJECTIVESLEARNING OBJECTIVES (continued)(continued)1010

After studying this chapter, you should be able to:After studying this chapter, you should be able to:

10-6

10-1 Using Statistics10-1 Using Statistics

• RegressionRegression refers to the statistical technique of modeling the relationship between variables.• In simple linearsimple linear regressionregression, we model the relationship

between two variablestwo variables. • One of the variables, denoted by Y, is called the dependent dependent

variable variable and the other, denoted by X, is called the independent variableindependent variable.

• The model we will use to depict the relationship between X and Y will be a straight-line relationshipstraight-line relationship.

• A graphical sketch of the the pairs (X, Y) is called a scatter scatter plotplot.

10-7

This scatterplot locates pairs of observations of advertising expenditures on the x-axis and sales on the y-axis. We notice that:

Larger (smaller) values of sales tend to be associated with larger (smaller) values of advertising.

Scatterplot of Advertising Expenditures (X) and Sales (Y)

50403020100

140

120

100

80

60

40

20

0

Advertising

Sa

les

The scatter of points tends to be distributed around a positively sloped straight line.

The pairs of values of advertising expenditures and sales are not located exactly on a straight line.

The scatter plot reveals a more or less strong tendency rather than a precise linear relationship.

The line represents the nature of the relationship on average.

10-1 Using Statistics10-1 Using Statistics

10-8

X

Y

X

Y

X 0

0

0

0

0

Y

X

Y

X

Y

XY

Examples of Other ScatterplotsExamples of Other Scatterplots

10-9

The inexact nature of the relationship between advertising and sales suggests that a statistical statistical modelmodel might be useful in analyzing the relationship.

A statistical model separates the systematic systematic componentcomponent of a relationship from the random componentrandom component.

The inexact nature of the relationship between advertising and sales suggests that a statistical statistical modelmodel might be useful in analyzing the relationship.

A statistical model separates the systematic systematic componentcomponent of a relationship from the random componentrandom component.

DataData

Statistical Statistical modelmodel

Systematic Systematic componentcomponent

++RandomRandomerrorserrors

In ANOVA, the systematic component is the variation of means between samples or treatments (SSTR) and the random component is the unexplained variation (SSE).

In regressionregression, the systematic component is the overall linear relationship, and the random component is the variation around the line.

In ANOVA, the systematic component is the variation of means between samples or treatments (SSTR) and the random component is the unexplained variation (SSE).

In regressionregression, the systematic component is the overall linear relationship, and the random component is the variation around the line.

Model BuildingModel Building

10-10

The population simple linear regression model:

Y= 0 + 1 X + Nonrandom or Random

Systematic Component Component

where Y is the dependent variable, the variable we wish to explain or predict X is the independent variable, also called the predictor variable is the error term, the only random component in the model, and thus, the only source of randomness in Y.

0 is the intercept of the systematic component of the regression relationship.1 is the slope of the systematic component.

The conditional mean of Y:

The population simple linear regression model:

Y= 0 + 1 X + Nonrandom or Random

Systematic Component Component

where Y is the dependent variable, the variable we wish to explain or predict X is the independent variable, also called the predictor variable is the error term, the only random component in the model, and thus, the only source of randomness in Y.

0 is the intercept of the systematic component of the regression relationship.1 is the slope of the systematic component.

The conditional mean of Y: E Y X X[ ] 0 1

10-2 The Simple Linear Regression 10-2 The Simple Linear Regression ModelModel

10-11

The simple linear regression model gives an exact linear relationship between the expected or average value of Y, the dependent variable, and X, the independent or predictor variable: E[Yi]=0 + 1 Xi

Actual observed values of Y differ from the expected value by an unexplained or random error:

Yi = E[Yi] + i

= 0 + 1 Xi + i

The simple linear regression model gives an exact linear relationship between the expected or average value of Y, the dependent variable, and X, the independent or predictor variable: E[Yi]=0 + 1 Xi

Actual observed values of Y differ from the expected value by an unexplained or random error:

Yi = E[Yi] + i

= 0 + 1 Xi + i

X

Y

E[Y]=0 + 1 X

Xi

}} 1 = Slope

1

0 = Intercept

Yi

{Error: i

Regression Plot

Picturing the Simple LinearPicturing the Simple Linear Regression Model Regression Model

10-12

• The relationship between X and Y is a straight-line relationship.

• The values of the independent variable X are assumed fixed (not random); the only randomness in the values of Y comes from the error term i.

• The errors i are normally distributed with mean 0 and variance 2. The errors are uncorrelated (not related) in successive observations. That is: ~ N(0,2)

• The relationship between X and Y is a straight-line relationship.

• The values of the independent variable X are assumed fixed (not random); the only randomness in the values of Y comes from the error term i.

• The errors i are normally distributed with mean 0 and variance 2. The errors are uncorrelated (not related) in successive observations. That is: ~ N(0,2) X

Y

E[Y]=0 + 1 X

Assumptions of the Simple Linear Regression Model

Identical normal distributions of errors, all centered on the regression line.

Assumptions of the Simple Linear Assumptions of the Simple Linear Regression ModelRegression Model

10-13

Estimation of a simple linear regression relationship involves finding estimated or predicted values of the intercept and slope of the linear regression line.

The estimated regression equation: Y = b0 + b1X + e

where b0 estimates the intercept of the population regression line, 0 ;b1 estimates the slope of the population regression line, 1;and e stands for the observed errors - the residuals from fitting the estimated regression line b0 + b1X to a set of n points.

Estimation of a simple linear regression relationship involves finding estimated or predicted values of the intercept and slope of the linear regression line.

The estimated regression equation: Y = b0 + b1X + e

where b0 estimates the intercept of the population regression line, 0 ;b1 estimates the slope of the population regression line, 1;and e stands for the observed errors - the residuals from fitting the estimated regression line b0 + b1X to a set of n points. The estimated regression line:

+

where Y (Y - hat) is the value of Y lying on the fitted regression line for a givenvalue of X.

Y b b X 0 1

The estimated regression line:

+

where Y (Y - hat) is the value of Y lying on the fitted regression line for a givenvalue of X.

Y b b X 0 1

10-3 Estimation: The Method of Least 10-3 Estimation: The Method of Least SquaresSquares

10-14

Fitting a Regression LineFitting a Regression Line

X

Y

Data

X

Y

Three errors from a fitted line

X

Y

Three errors from the least squares regression line

X

Errors from the least squares regression line are minimized

10-15

.{Error ei Yi Yi

Yi the predicted value of Y for Xi

Yi the predicted value of Y for Xi

YY

XX

Y b b X 0 1 the fitted regression lineY b b X 0 1 the fitted regression line

Yi

Yi

Errors in RegressionErrors in Regression

XXii

point data observed the point data observed the

10-16

Least Squares RegressionLeast Squares Regression

The sum of squared errors in regression is:

SSE = e (y

The is that which the SSEwith respect to the estimates b and b .

The :

y x

x y x x

i

2

i=1

n

ii=1

n

0 1

ii=1

n

ii=1

n

i ii=1

n

ii=1

n

i

2

i=1

n

)y

nb b

b b

i

2

0 1

0 1

least squares regression line

normal equations

minimizes

b0SSE

b1

Least squares b0

Least squares b1

At this point SSE is minimized with respect to b0 and b1

10-17

Sums of Squares and Cross Products:

Least squares regression estimators:

SS x x xx

n

SS y y yy

n

SS x x y y xyx y

n

bSSSS

b y b x

x

y

xy

XY

X

( )

( )

( )( )( )

2 2

2

2 2

2

1

0 1

Sums of Squares and Cross Products:

Least squares regression estimators:

SS x x xx

n

SS y y yy

n

SS x x y y xyx y

n

bSSSS

b y b x

x

y

xy

XY

X

( )

( )

( )( )( )

2 2

2

2 2

2

1

0 1

Sums of Squares, Cross Products, Sums of Squares, Cross Products, and Least Squares Estimatorsand Least Squares Estimators

10-18

Miles Dollars Miles 2 Miles*Dollars 1211 1802 1466521 2182222 1345 2405 1809025 3234725 1422 2005 2022084 2851110 1687 2511 2845969 4236057 1849 2332 3418801 4311868 2026 2305 4104676 4669930 2133 3016 4549689 6433128 2253 3385 5076009 7626405 2400 3090 5760000 7416000 2468 3694 6091024 9116792 2699 3371 7284601 9098329 2806 3998 7873636 11218388 3082 3555 9498724 10956510 3209 4692 10297681 15056628 3466 4244 12013156 14709704 3643 5298 13271449 19300614 3852 4801 14837904 18493452 4033 5147 16265089 20757852 4267 5738 18207288 24484046 4498 6420 20232004 28877160 4533 6059 20548088 27465448 4804 6426 23078416 30870504 5090 6321 25908100 32173890 5233 7026 27384288 36767056 5439 6964 29582720 3787719679,448 106,605 293,426,946 390,185,014

Miles Dollars Miles 2 Miles*Dollars 1211 1802 1466521 2182222 1345 2405 1809025 3234725 1422 2005 2022084 2851110 1687 2511 2845969 4236057 1849 2332 3418801 4311868 2026 2305 4104676 4669930 2133 3016 4549689 6433128 2253 3385 5076009 7626405 2400 3090 5760000 7416000 2468 3694 6091024 9116792 2699 3371 7284601 9098329 2806 3998 7873636 11218388 3082 3555 9498724 10956510 3209 4692 10297681 15056628 3466 4244 12013156 14709704 3643 5298 13271449 19300614 3852 4801 14837904 18493452 4033 5147 16265089 20757852 4267 5738 18207288 24484046 4498 6420 20232004 28877160 4533 6059 20548088 27465448 4804 6426 23078416 30870504 5090 6321 25908100 32173890 5233 7026 27384288 36767056 5439 6964 29582720 3787719679,448 106,605 293,426,946 390,185,014

85.274

25

448,79)255333776.1(

25

605,106

10

26.1255333776.184.557,947,40

4.852,402,51

1

4.852,402,5125

)605,106)(448,79(014,185,390

)(

84.557,947,4025

2448,79946,426,293

22

xbyb

XSS

XYSS

b

n

yxxyxySS

n

xxxSS

85.274

25

448,79)255333776.1(

25

605,106

10

26.1255333776.184.557,947,40

4.852,402,51

1

4.852,402,5125

)605,106)(448,79(014,185,390

)(

84.557,947,4025

2448,79946,426,293

22

xbyb

XSS

XYSS

b

n

yxxyxySS

n

xxxSS

Example 10-1Example 10-1

10-19

Template (partial output) that can be Template (partial output) that can be used to carry out a Simple Regressionused to carry out a Simple Regression

10-20

Template (continued) that can be used Template (continued) that can be used to carry out a Simple Regressionto carry out a Simple Regression

10-21


Residual Analysis. The plot shows the absence of a relationshipbetween the residuals and the X-values (miles).Residual Analysis. The plot shows the absence of a relationshipbetween the residuals and the X-values (miles).

10-22


Note:Note: The normal probability plot is approximately linear. This would indicate that the normality assumption for the errors has not been violated.

Note:Note: The normal probability plot is approximately linear. This would indicate that the normality assumption for the errors has not been violated.

10-23

Y

X

What you see when looking at the total variation of Y.

X

What you see when looking along the regression line at the error variance of Y.

Y

Total Variance and Error VarianceTotal Variance and Error Variance

10-24

Degrees of Freedom in Regression:

An unbiased estimator of s2

, denoted by S2

:

df = (n - 2) (n total observations less one degree of freedom

for each parameter estimated (b0 and b1) )

= ( - )

=

MSE =SSE

(n - 2)

SSE Y Y SSY

SS XY

SS XSSY b SS XY

( )2

2

1

X

Y

Square and sum all regression errors to find SSE.

Example 10 - 1:

SSE SSY b SS XY

MSESSE

n

s MSE

=

166855898 1 255333776 51402852 4

2328161 2

2

2328161 2

23101224 4

101224 4 318 158

( . )( . )

.

.

.

. .

10-4 Error Variance and the Standard 10-4 Error Variance and the Standard Errors of Regression EstimatorsErrors of Regression Estimators

10-25

The standard error of (intercept)

where s = MSE

The standard error of (slope)

0

1

b

s bs x

nSS

b

s bs

SS

X

X

:

( )

:

( )

0

2

1

The standard error of (intercept)

where s = MSE

The standard error of (slope)

0

1

b

s bs x

nSS

b

s bs

SS

X

X

:

( )

:

( )

0

2

1

Example 10 - 1:

s bs x

nSS X

s bs

SS X

( )

.

( )( . ).

( )

.

..

0

2

318 158 293426944

25 4097557 84170 338

1

318 158

40947557 840 04972

Example 10 - 1:

s bs x

nSS X

s bs

SS X

( )

.

( )( . ).

( )

.

..

0

2

318 158 293426944

25 4097557 84170 338

1

318 158

40947557 840 04972

Standard Errors of Estimates in Standard Errors of Estimates in RegressionRegression

10-26

A (1 - ) 100% confidence interval for b0

A (1 - ) 100% confidence interval for b1

:

,( )( )

:

,( )( )

b tn

s b

b tn

s b

02

2 0

12

2 1

Example 10 - 195% Confidence Intervals:b t s b

b t s b

0 0 025 25 2 0

0 025 25 2

170 33827485 352 43

7758 627 28

01 25533 010287115246 1 35820

1 1

. ,( ) ( )

. ,( ) ( )

( . ). .

[ . , . ]

( ). .

[ . , . ]

= 274.85 2.069) (

= 1.25533 2.069) ( .04972

Example 10 - 195% Confidence Intervals:b t s b

b t s b

0 0 025 25 2 0

0 025 25 2

170 33827485 352 43

7758 627 28

01 25533 010287115246 1 35820

1 1

. ,( ) ( )

. ,( ) ( )

( . ). .

[ . , . ]

( ). .

[ . , . ]

= 274.85 2.069) (

= 1.25533 2.069) ( .04972

Length = 1H

eight = Slope

Least-squares point estimate:b1=1.25533

Upper

95%

bou

nd o

n slo

pe: 1

.358

20

Lower 95% bound: 1

.15246

(not a possible value of the regression slope at 95%)

0

Confidence Intervals for the Confidence Intervals for the Regression ParametersRegression Parameters

10-27

Template (partial output) that can be used Template (partial output) that can be used to obtain Confidence Intervals for to obtain Confidence Intervals for and and

10-28

The correlationcorrelation between two random variables, X and Y, is a measure of the degree of linear associationdegree of linear association between the two variables.

The population correlation, denoted by, can take on any value from -1 to 1.

The correlationcorrelation between two random variables, X and Y, is a measure of the degree of linear associationdegree of linear association between the two variables.

The population correlation, denoted by, can take on any value from -1 to 1.

indicates a perfect negative linear relationship-1 < < 0 indicates a negative linear relationship indicates no linear relationship0 < < 1 indicates a positive linear relationshipindicates a perfect positive linear relationship

The absolute value of indicates the strength or exactness of the relationship.

indicates a perfect negative linear relationship-1 < < 0 indicates a negative linear relationship indicates no linear relationship0 < < 1 indicates a positive linear relationshipindicates a perfect positive linear relationship

The absolute value of indicates the strength or exactness of the relationship.

10-5 Correlation10-5 Correlation

10-29

Y

X

= 0= 0

Y

X

= -.8= -.8 Y

X

= .8= .8

Y

X

= 0= 0

Y

X

= -1= -1Y

X

= 1= 1

Illustrations of CorrelationIllustrations of Correlation

10-30

The sample correlation coefficient*:

=rSS

XYSS

XSS

Y

The population correlation coefficient:

=

Cov X Y

X Y

( , )

The covariance of two random variables X and Y: where X and are the population means of X and Y respectivelyY .

Cov X Y E X X Y Y( , ) [( )( )]

Example 10 - 1:

=

rSS

XYSS

XSS

Y

51402852.4

40947557.84 6685589851402852.4

52321943 299824

( )( )

..

*Note:Note: If < 0, b1 < 0 If = 0, b1 = 0 If > 0, b1 >0*Note:Note: If < 0, b1 < 0 If = 0, b1 = 0 If > 0, b1 >0

Covariance and CorrelationCovariance and Correlation

10-31

H0: = 0 (No linear relationship)H1: 0 (Some linear relationship)

Test Statistic: tr

rn

n( )

2 212

Example 10 -1:

=0.98241- 0.9651

25- 2

=0.98240.0389

H rejected at 1% level0

tr

rn

t

n( )

.

.

. .

2 2

0 005

12

2525

2 807 2525

Example 10 -1:

=0.98241- 0.9651

25- 2

=0.98240.0389

H rejected at 1% level0

tr

rn

t

n( )

.

.

. .

2 2

0 005

12

2525

2 807 2525

Hypothesis Tests for the Correlation Hypothesis Tests for the Correlation CoefficientCoefficient

10-32

Y

X

Y

X

Y

X

Constant Y Unsystematic Variation Nonlinear Relationship

A hypothesis test for the existence of a linear relationship between X and Y:

H 0 H1Test statistic for the existence of a linear relationship between X and Y:

( - )

where is the least - squares estimate of the regression slope and ( ) is the standard error of .

When the null hypothesis is true, the statistic has a distribution with - degrees of freedom.

:

:

( )

1 0

1 0

2

1

1

1 1 12

tn

b

s b

b s b b

t n

10-6 Hypothesis Tests about the 10-6 Hypothesis Tests about the Regression RelationshipRegression Relationship

10-33

Example 10 - 1:

H 0 H1

=1.25533

0.04972

H 0 is rejected at the 1% level and we may

conclude that there is a relationship between

charges and miles traveled.

( - )

:

:

( )

.

. .( . , )

1 0

1 0

1

1

25 25

2 807 25 25

2

0 005 23

t

b

s b

t

n

1. fromdifferent ist coefficien

beta that theconcludenot may We

level. 10% at the rejectednot is 0

H

14.1671.1)58,05.0(

14.10.21

1-1.24=

)1

(

11

)2-(

11

:1

H

11

:0

H

:4-10 Example

t

bs

b

nt

Hypothesis Tests for the Regression Hypothesis Tests for the Regression SlopeSlope

10-34

The coefficient of determination, rcoefficient of determination, r22, is a descriptive measure of the strength of the regression relationship, a measure of how well the regression line fits the data.

.{

Y

X

Y

Y

Y

X

{}Total DeviationTotal Deviation

Explained DeviationExplained Deviation

Unexplained DeviationUnexplained Deviation

Total = Unexplained ExplainedDeviation Deviation Deviation (Error) (Regression)

SST = SSE + SSR

r2

( ) ( ) ( )

( ) ( ) ( )

y y y y y y

y y y y y y

SSR

SST

SSE

SST

2 2 2

1Percentage of total variation explained by the regression.

Percentage of total variation explained by the regression.

10-7 How Good is the Regression?10-7 How Good is the Regression?

10-35

Y

X

r2 = 0 SSE

SST

Y

X

r2 = 0.90SSE

SST

SSR

Y

X

r2 = 0.50 SSE

SST

SSR

Example 10 -1:

r 2 SSRSST

64527736 866855898

0 96518.

.

5500500045004000350030002500200015001000

7000

6000

5000

4000

3000

2000

Miles

Dol

lar s

The Coefficient of DeterminationThe Coefficient of Determination

10-36

10-8 Analysis-of-Variance Table and 10-8 Analysis-of-Variance Table and an an FF Test of the Regression Model Test of the Regression Model

Example 10-1

Source ofVariation

Sum ofSquares

Degrees ofFreedom

Mean SquareF Ratio p Value

Regression 64527736.8 1 64527736.8 637.47 0.000

Error 2328161.2 23 101224.4

Total 66855898.0 24

Example 10-1

Source ofVariation

Sum ofSquares

Degrees ofFreedom

Mean SquareF Ratio p Value

Regression 64527736.8 1 64527736.8 637.47 0.000

Error 2328161.2 23 101224.4

Total 66855898.0 24

Source ofVariation

Sum ofSquares

Degrees ofFreedom Mean Square F Ratio

Regression SSR (1) MSR MSRMSE

Error SSE (n-2) MSE

Total SST (n-1) MST

Source ofVariation

Sum ofSquares

Degrees ofFreedom Mean Square F Ratio

Regression SSR (1) MSR MSRMSE

Error SSE (n-2) MSE

Total SST (n-1) MST

10-37

Template (partial output) that displays Analysis of Template (partial output) that displays Analysis of Variance and an Variance and an FF Test of the Regression Model Test of the Regression Model

10-38

x or y

0

Residuals

Homoscedasticity: Residuals appear completely random. No indication of model inadequacy.

0

Residuals

Curved pattern in residuals resulting from underlying nonlinear relationship.

0

Residuals

Residuals exhibit a linear trend with time.

Time

0

Residuals

Heteroscedasticity: Variance of residuals increases when x changes.

x or y

x or y

10-9 Residual Analysis and Checking 10-9 Residual Analysis and Checking for Model Inadequaciesfor Model Inadequacies

10-39

Normal Probability Plot of the Normal Probability Plot of the ResidualsResiduals

Flatter than NormalFlatter than Normal

10-40


More Peaked than NormalMore Peaked than Normal

10-41


Positively Skewed Positively Skewed

10-42


Negatively Skewed Negatively Skewed

10-43

• Point Prediction A single-valued estimate of Y for a given value of X

obtained by inserting the value of X in the estimated regression equation.

• Prediction Interval For a value of Y given a value of X

Variation in regression line estimate Variation of points around regression line

For an average value of Y given a value of X Variation in regression line estimate

• Point Prediction A single-valued estimate of Y for a given value of X

obtained by inserting the value of X in the estimated regression equation.

• Prediction Interval For a value of Y given a value of X

Variation in regression line estimate Variation of points around regression line

For an average value of Y given a value of X Variation in regression line estimate

10-10 Use of the Regression Model 10-10 Use of the Regression Model for Predictionfor Prediction

10-44

X

Y

X

Y

Regression line

Upper limit on slope

Lower limit on slope

1) Uncertainty about the slope of the regression line

X

Y

X

Y

Regression lineUpper limit on intercept

Lower limit on intercept

2) Uncertainty about the intercept of the regression line

Errors in Predicting Errors in Predicting E[Y|X]E[Y|X]

10-45

X

Y

X

Prediction Interval for E[Y|X]

Y

Regression line

• The prediction band for E[Y|X] is narrowest at the mean value of X.

• The prediction band widens as the distance from the mean of X increases.

• Predictions become very unreliable when we extrapolate beyond the range of the sample itself.

• The prediction band for E[Y|X] is narrowest at the mean value of X.

• The prediction band widens as the distance from the mean of X increases.

• Predictions become very unreliable when we extrapolate beyond the range of the sample itself.

Prediction Interval for Prediction Interval for E[Y|X]E[Y|X]

Prediction band for E[Y|X]

10-46

Additional Error in Predicting Individual Additional Error in Predicting Individual Value of Value of YY

3) Variation around the regression line

X

YRegression line

X

Y

X

Prediction Interval for E[Y|X]

Y

Regression line

Prediction band for E[Y|X]

Prediction band for Y

10-47

]67.5972 ,43.4619[62.67605.5296

84.557,947,40

)92.177,3000,4(

25

1116.318069.2 ,000)}(1.2553)(474.852{

:4,000)=(X 1-10 Example

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10-48

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10-49

Template Output with Prediction Template Output with Prediction IntervalsIntervals

10-50

10-11 The Solver Method for 10-11 The Solver Method for RegressionRegression

The solver macro available in EXCEL can also be used to conduct a simple linear regression. See the text for instructions.See the text for instructions.

10-51

• The Case of Independent Random Variables: For independent random variables, X1, X2, …, Xn, the

expected value for the sum, is given by:• E(X1 + X2 + … + Xn) = E(X1) + E(X2)+ … + E(Xn)

• For independent random variables, X1, X2, …, Xn, the variance for the sum, is given by:

• V(X1 + X2 + … + Xn) = V(X1) + V(X2)+ … + V(Xn)

• The Case of Independent Random Variables: For independent random variables, X1, X2, …, Xn, the

expected value for the sum, is given by:• E(X1 + X2 + … + Xn) = E(X1) + E(X2)+ … + E(Xn)

• For independent random variables, X1, X2, …, Xn, the variance for the sum, is given by:

• V(X1 + X2 + … + Xn) = V(X1) + V(X2)+ … + V(Xn)

10-12 Linear Composites of 10-12 Linear Composites of Dependent Random VariablesDependent Random Variables

10-52

• The Case of Independent Random Variables with Weights: For independent random variables, X1, X2, …, Xn, with

respective weights 1, 2, …, n, the expected value for the sum, is given by:

• E(1 X1 + 2 X2 + … + n Xn) = 1 E(X1) + 2 E(X2)+ … + n E(Xn) For independent random variables, X1, X2, …, Xn, with

respective weights 1, 2, …, n, the variance for the sum, is given by:

• V(1 X1 + 2 X2 + … + n Xn) = 12 V(X1) + 2

2 V(X2)+ … + n

2 V(Xn)

• The Case of Independent Random Variables with Weights: For independent random variables, X1, X2, …, Xn, with

respective weights 1, 2, …, n, the expected value for the sum, is given by:

• E(1 X1 + 2 X2 + … + n Xn) = 1 E(X1) + 2 E(X2)+ … + n E(Xn) For independent random variables, X1, X2, …, Xn, with


• V(1 X1 + 2 X2 + … + n Xn) = 12 V(X1) + 2

2 V(X2)+ … + n

2 V(Xn)


10-53

• The covariance between two random variables X1 and X2 is given by:

• Cov(X1, X2) = E{[X1 – E(X1)] [X2 – E(X2)]}

• A simpler measure of covariance is given by:

• Cov(X1, X2) = SD(X1) SD(X2) where is the correlation between X1 and X2.

• The covariance between two random variables X1 and X2 is given by:

• Cov(X1, X2) = E{[X1 – E(X1)] [X2 – E(X2)]}

• A simpler measure of covariance is given by:

• Cov(X1, X2) = SD(X1) SD(X2) where is the correlation between X1 and X2.

CovarianceCovariance of two random variables of two random variables XX11 and X and X22

10-54

• The Case of Dependent Random Variables with Weights: For dependent random variables, X1, X2, …, Xn, with


• V(1 X1 + 1 X2 + … + n Xn) = 12 V(X1) + 2

2 V(X2)+ … + n

2 V(Xn) + 2 1 2Cov(X1, X2) + … + 2 n-1 nCov(Xn-1, Xn)

• The Case of Dependent Random Variables with Weights: For dependent random variables, X1, X2, …, Xn, with


• V(1 X1 + 1 X2 + … + n Xn) = 12 V(X1) + 2

2 V(X2)+ … + n

2 V(Xn) + 2 1 2Cov(X1, X2) + … + 2 n-1 nCov(Xn-1, Xn)


Documents

10-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)