linear regression 12
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8/3/2019 Linear Regression 12
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2003 Prentice-Hall, Inc. Chap 10-1
Business Statistics:A First Course
(3rd Edition)
Chapter 10
Simple Linear Regression
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8/3/2019 Linear Regression 12
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2003 Prentice-Hall, Inc.Chap 10-2
Chapter Topics
Types of Regression Models
Determining the Simple Linear Regression
Equation Measures of Variation
Assumptions of Regression and Correlation
Residual Analysis Measuring Autocorrelation
Inferences about the Slope
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8/3/2019 Linear Regression 12
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2003 Prentice-Hall, Inc.Chap 10-3
Chapter Topics
Correlation - Measuring the Strength of theAssociation
Estimation of Mean Values and Prediction ofIndividual Values
Pitfalls in Regression and Ethical Issues
(continued)
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8/3/2019 Linear Regression 12
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2003 Prentice-Hall, Inc.Chap 10-4
Purpose of Regression Analysis
Regression Analysis is Used Primarily to ModelCausality and Provide Prediction
Predict the values of a dependent (response)variable based on values of at least oneindependent (explanatory) variable
Explain the effect of the independent variables on
the dependent variable
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2003 Prentice-Hall, Inc.Chap 10-5
Types of Regression Models
Positive Linear Relationship
Negative Linear Relationship
Relationship NOT Linear
No Relationship
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2003 Prentice-Hall, Inc.Chap 10-6
Simple Linear Regression Model
Relationship Between Variables is Describedby a Linear Function
The Change of One Variable Causes the OtherVariable to Change
A Dependency of One Variable on the Other
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8/3/2019 Linear Regression 12
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Chap 10-7
PopulationRegressionLine
(conditional mean)
Simple Linear Regression Model
Population regression line is a straight line thatdescribes the dependence of the average valueaverage value
(conditional mean)(conditional mean) of one variable on the other
PopulationY intercept
PopulationSlopeCoefficient
RandomError
Dependent(Response)Variable
Independent(Explanatory)Variable
ii iY X 0 1+ +=|Y X
(continued)
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8/3/2019 Linear Regression 12
8/65 2003 Prentice-Hall, Inc.Chap 10-8
Simple Linear Regression Model(continued)
ii iY X
0 1+ +=
= Random Error
Y
X
(Observed Value ofY) =
Observed Value ofY
|Y X iX 0 1= +
i
0
1
(Conditional Mean)
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8/3/2019 Linear Regression 12
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Sample regression line provides an estimateestimateofthe population regression line as well as apredicted value of Y
Linear Regression Equation
Sample
Y Intercept
Sample
Slope
Coefficient
Residual0 1i iib bY X e+ +=
0 1Y b b X = + =
Simple Regression Equation
(Fitted Regression Line, Predicted Value)
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8/3/2019 Linear Regression 12
10/65 2003 Prentice-Hall, Inc.Chap 10-10
Linear Regression Equation
and are obtained by finding the valuesof and that minimizes the sum of the
squared residuals
provides an estimateestimateof
provides and estimateestimateof
0b 1b
0b 1b
0b 01b 1
(continued)
( )2
2
1 1
n n
i i i
i i
Y Y e= =
=
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Linear Regression Equation(continued)
Y
X
Observed Value
|Y X iX 0 1= +
i
0
1
ii iY X 0 1+ +=
0 1
i iY b b X = +
ie
0 1i iib bY X e+ +=
1b
0b
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8/3/2019 Linear Regression 12
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Interpretation of the Slope andIntercept
is the average value of Y when
the value of X is zero.
measures the change in the
average value of Y as a result of a one-unitchange in X.
| 0Y X 0 ==
|
1
Y X
X
=
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8/3/2019 Linear Regression 12
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Interpretation of the Slope and
Intercept
is the estimatedestimatedaverage value
of Y when the value of X is zero.
is the estimatedestimatedchange in the
average value of Y as a result of a one-unit
change in X.
(continued)
| 0
Y Xb
0 ==
|
1
Y X
bX
=
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8/3/2019 Linear Regression 12
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Simple Linear Regression:Example
You wish to examinethe linear dependency
of the annual sales ofproduce stores on theirsizes in square footage.Sample data for 7
stores were obtained.Find the equation ofthe straight line thatfits the data best.
AnnualStore Square Sales
Feet ($1000)
1 1,726 3,6812 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
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Scatter Diagram: Example
0
2000
4000
6000
8000
10000
12000
0 1000 2000 3000 4000 5000 6000
Sq u a re F e e t
A
nnualSales
($000)
Excel Output
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8/3/2019 Linear Regression 12
16/65 2003 Prentice-Hall, Inc.Chap 10-16
Simple Linear RegressionEquation: Example
0 1
1636.415 1.487
i i
i
Y b b X
X
= +
= +
From Excel Printout:
Coefficients
Intercept 1636.414726
X Variable 1.486633657
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8/3/2019 Linear Regression 12
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Graph of the Simple LinearRegression Equation: Example
0
2000
4000
6000
8000
10000
12000
0 1000 2000 3000 4000 5000 6000
Square Fee t
A
nnualSales
($000)
Yi=16
36.415
+1.48
7Xi
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Chap 10-18
Interpretation of Results:Example
The slope of 1.487 means that each increase of oneunit in X, we predict the average of Y to increase byan estimated 1.487 units.
The equation estimates that foreach increase of 1square footin the size of the store, the expectedannual sales are predictedto increase by $1487.
1636.415 1.487i iY X= +
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Chap 10-19
Simple Linear Regression inPHStat
In Excel, use PHStat | Regression | Simple
Linear Regression
EXCEL Spreadsheet of Regression Sales onFootage
Microsoft Excel
Worksheet
f
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Chap 10-20
Measures of Variation:
The Sum of Squares
SST = SSR + SSE
Total
Sample
Variability
= Explained
Variability
+ Unexplained
Variability
f i i
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Chap 10-21
Measures of Variation:
The Sum of Squares
SST = Total Sum of Squares Measures the variation of the Yi values around
their mean,
SSR = Regression Sum of Squares Explained variation attributable to the relationship
between Xand Y
SSE = Error Sum of Squares Variation attributable to factors other than the
relationship between Xand Y
(continued)
Y
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Chap 10-22
Measures of Variation:The Sum of Squares
(continued)
Xi
Y
X
Y
SST=(Yi
-Y)2
SSE=(Yi-Yi)2
SSR= (Yi-Y)2
_
_
_
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Chap 10-23
The ANOVA Table in Excel
ANOVAdf SS MS F Significance
FRegression k SSR MSR =SSR/k
MSR/MSE P-value of
the F Test
Residuals n-k-1 SSE MSE=SSE/(n-k-1)Total n-1 SST
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2003 Prentice-Hall, Inc.Chap 10-24
Measures of VariationThe Sum of Squares: Example
A NO V A
d f SS MS F Signi ficance F
Regress ion 1 3 0380 456 .12 303 8045 6 8 1.17 909 0.00 028 120
Residual 5 1 87 11 99 .5 95 3 74 23 9.9 2
Total 6 32251655.71
Excel Output for Produce Stores
SSR
SSERegression (explained) df
Degrees of freedom
Error (residual) df
Total df
SST
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2003 Prentice-Hall, Inc.Chap 10-25
The Coefficient of Determination
Measures the proportion of variation in Y thatis explained by the independent variable X inthe regression model
2 Regression Sum of Squares
Total Sum of Squares
SSRr
SST= =
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2003 Prentice-Hall, Inc.Chap 10-26
Coefficients of Determination (r2)and Correlation (r)
r2 = 1, r2 = 1,
r2 = .81, r2 = 0,Y
Yi= b0 + b1XiX
^
Y
Yi= b0 + b1Xi
X
^Y
Yi= b0 + b1XiX
^
Y
Yi= b0 + b1XiX
^
r= +1 r= -1
r= +0.9 r= 0
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2003 Prentice-Hall, Inc.Chap 10-27
Standard Error of Estimate
The standard deviation of the variation ofobservations around the regression equation
( )2
1
2 2
n
i
i
YX
Y YSSE
Sn n
=
= =
f
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2003 Prentice-Hall, Inc.Chap 10-28
Measures of Variation:Produce Store Example
Regression Statistics
Multiple R 0.9705572
R Square 0.94198129
Adjusted R Square 0.93037754
Standard Error 611.751517
Observations 7
Excel Output for Produce Stores
r2 = .94
94% of the variation in annual sales can beexplained by the variability in the size of the
store as measured by square footage
Syx
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2003 Prentice-Hall, Inc.Chap 10-29
Linear Regression Assumptions
Normality Y values are normally distributed for each X
Probability distribution of error is normal 2. Homoscedasticity (Constant Variance)
3. Independence of Errors
V i ti f E A d th
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2003 Prentice-Hall, Inc.Chap 10-30
Y values are normally distributed
around the regression line.
For eachXvalue, the spread or
variance around the regression line isthe same.
Variation of Errors Around theRegression Line
X1
X2
X
Y
f()
Sample Regression Line
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2003 Prentice-Hall, Inc.Chap 10-31
Residual Analysis
Purposes Examine linearity
Evaluate violations of assumptions Graphical Analysis of Residuals
Plot residuals vs. X and time
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2003 Prentice-Hall, Inc.Chap 10-32
Residual Analysis for Linearity
Not Linear Linear
X
e eX
Y
X
Y
X
R id l A l i f
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2003 Prentice-Hall, Inc.Chap 10-33
Residual Analysis forHomoscedasticity
Heteroscedasticity Homoscedasticity
SR
X
SR
X
Y
X X
Y
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2003 Prentice-Hall, Inc.Chap 10-34
Residual Plot
0 1000 2000 3000 4000 5000 6000
Square Feet
Residual Analysis:Excel Outputfor Produce Stores Example
Excel Output
Observation Predicted Y Residuals1 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 644.09816037 3588.364717 171.6352829
f
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2003 Prentice-Hall, Inc.Chap 10-35
Residual Analysis forIndependence
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
2
1
2
2
1
( )n
i i
i
n
i
i
e e
D
e
=
=
=
Should be close to 2.If not, examine the modelfor autocorrelation.
D bi W t St ti ti i
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2003 Prentice-Hall, Inc.Chap 10-36
Durbin-Watson Statistic inPHStat
PHStat | Regression | Simple LinearRegression Check the box for Durbin-Watson Statistic
Obt i i th C iti l V l f
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2003 Prentice-Hall, Inc.Chap 10-37
5 = .
k=1 k=2
n dL dU dL dU
15 1.08 1.36 .95 1.54
16 1.10 1.37 .98 1.54
Obtaining the Critical Values ofDurbin-Watson Statistic
Table 13.4 Finding critical values of Durbin-Watson Statistic
U i th D bi W t
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2003 Prentice-Hall, Inc.Chap 10-38
Accept H0(no autocorrelatin)
Using the Durbin-WatsonStatistic
: No autocorrelation (error terms are independent)
: There is autocorrelation (error terms are notindependent)
0H
1H
0 42dL
4-dL
dU
4-dU
Reject H0(positive
autocorrelation)
Inconclusive Reject H0(negative
autocorrelation)
Resid al Anal sis fo
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2003 Prentice-Hall, Inc.Chap 10-39
Residual Analysis forIndependence
Not Independent Independente e
TimeTime
Residual Is Plotted Against Time to Detect Any Autocorrelation
No Particular PatternCyclical Pattern
Graphical Approach
Inference about the Slope:
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2003 Prentice-Hall, Inc.Chap 10-40
Inference about the Slope:t Test
t Test for a Population Slope Is there a linear dependency ofYon X?
Null and Alternative Hypotheses
H0: 1 = 0 (No Linear Dependency) H1: 1 0 (Linear Dependency)
Test Statistic
1
1
1 1
2
1
where
( )
YX
bn
b
i
i
b St SS
X X
=
= =
. . 2d f n=
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2003 Prentice-Hall, Inc.Chap 10-41
Example: Produce Store
Data for 7 Stores:
Estimated RegressionEquation:
The slope of thismodel is 1.487.
Does Square FootageAffect Annual Sales?
AnnualStore Square Sales
Feet ($000)
1 1,726 3,681
2 1,542 3,395
3 2,816 6,653
4 5,555 9,5435 1,292 3,318
6 2,208 5,563
7 1,313 3,760
1636.415 1.487 iY X= +
Inferences about the Slope:
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2003 Prentice-Hall, Inc.Chap 10-42
Inferences about the Slope:t Test Example
H0: 1 = 0H1: 1 0= .05df= 7 - 2 = 5Critical Value(s):
Test Statistic:
Decision:Conclusion:
There is evidence that
square footage affects
annual sales.
t0 2.5706-2.5706
.025
Reject Reject
.025
From Excel Printout
Reject H0
Coef fic ientsS tandard Errort Stat P -value
In te rc e pt 1 63 6. 41 47 4 51 .4 95 3 3 .6 24 4 0 .0 15 1F ootage 1.4866 0.1650 9.0099 0.0002
1b 1bS t
Inferences about the Slope:
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2003 Prentice-Hall, Inc.Chap 10-43
Inferences about the Slope:Confidence Interval Example
Confidence Interval Estimate of the Slope:
11 2n bb t S
Excel Printout for Produce Stores
At 95% level of confidence the confidence interval forthe slope is (1.062, 1.911). Does not include 0.
Conclusion:There is a significant linear dependencyof annual sales on the size of the store.
Lower 95% Upper 95%
Intercept 475.810926 2797.01853
X Va riable 1.06249037 1.91077694
Inferences about the Slope:
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2003 Prentice-Hall, Inc.Chap 10-44
Inferences about the Slope:F Test
F Test for a Population Slope Is there a linear dependency ofYon X?
Null and Alternative Hypotheses
H0: 1 = 0 (No Linear Dependency) H1: 1 0 (Linear Dependency)
Test Statistic
Numerator d.f.=1, denominator d.f.=n-2
( )
1
2
SSR
FSSE
n
=
Relationship between a t Test
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2003 Prentice-Hall, Inc.Chap 10-45
Relationship between a t Testand an F Test
Null and Alternative Hypotheses H0: 1 = 0 (No Linear Dependency)
H1: 1 0 (Linear Dependency)
( )2
2 1, 2n nt F =
Inferences about the Slope:
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2003 Prentice-Hall, Inc.Chap 10-46
A N O V A
d f S S M S F S ig n i f i c a n
R e g r e ssi o n 1 3 0 3 8 0 4 5 6 . 1 23 0 3 8 0 4 5 6 . 1 28 1 .1 79 0 . 0 0 0 2
R e si d u a l 5 1 8 7 1 1 9 9 .5 9 53 7 4 2 3 9 . 9 1 9
T o ta l 6 3 2 2 5 1 6 5 5 . 7 1
Inferences about the Slope:F Test Example
Test Statistic:
Decision:
Conclusion:
H0: 1= 0H1: 1 0= .05numeratordf = 1denominatordf= 7 - 2 = 5
There is evidence that
square footage affects
annual sales.
From Excel Printout
Reject H0
0 6.61
Reject
= .05
1, 2nF
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2003 Prentice-Hall, Inc.Chap 10-47
Purpose of Correlation Analysis
Correlation Analysis is Used to MeasureStrength of Association (Linear Relationship)Between 2 Numerical Variables Only Strength of the Relationship is Concerned No Causal Effect is Implied
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2003 Prentice-Hall, Inc.Chap 10-48
Purpose of Correlation Analysis
Population Correlation Coefficient(Rho) isUsed to Measure the Strength between the
Variables
Sample Correlation Coefficient r is anEstimate of and is Used to Measure theStrength of the Linear Relationship in the
Sample Observations
(continued)
Sample of Observations from
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2003 Prentice-Hall, Inc.Chap 10-49r= .6 r= 1
Sample of Observations fromVarious r Values
Y
X
Y
X
Y
X
Y
X
Y
X
r= -1 r= -.6 r= 0
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2003 Prentice-Hall, Inc.Chap 10-50
Features of and r
Unit Free
Range between -1 and 1
The Closer to -1, the Stronger the NegativeLinear Relationship
The Closer to 1, the Stronger the PositiveLinear Relationship
The Closer to 0, the Weaker the LinearRelationship
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2003 Prentice-Hall, Inc.Chap 10-51
t Test for Correlation
Hypotheses H0:= 0 (No Correlation)
H1: 0 (Correlation)
Test Statistic
( ) ( )
( ) ( )
2
2 1
2 2
1 1
where
2
n
i i
i
n n
i i
i i
rt
r
n
X X Y Y
r r
X X Y Y
=
= =
=
1
= =
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2003 Prentice-Hall, Inc.Chap 10-52
Example: Produce Stores
Regression Statistics
M ult iple R 0.9705572
R S quare 0.94198129
A djus ted R S quare 0.93037754
S t andard E rror 611.751517
O bs ervations 7
From Excel Printout r
Is there anyevidence of linear
relationship betweenAnnual Sales of astore and its SquareFootage at .05 levelof significance? H0:= 0 (No association)
H1: 0 (Association)
= .05df= 7 - 2 = 5
Example: Produce Stores
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Example: Produce StoresSolution
0 2.5706-2.5706
.025Reject Reject.025
Critical Value(s):
Conclusion:There is evidence of a linear
relationship at 5% level of
significance
Decision:Reject H0
2
.97069.0099
1 .9420
52
rt
r
n
= = =
1
The value of the t statistic isexactly the same as the tstatistic value for test on theslope coefficient
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2003 Prentice-Hall, Inc.Chap 10-54
Estimation of Mean Values
Confidence Interval Estimate for :
The Mean ofYgiven a particular Xi
2
22
1
( )1
( )
ii n YX n
i
i
X XY t S n
X X
=
+
t value from table
with df=n-2
Standard error
of the estimate
Size of interval vary according todistance away from mean, X
| iY X X
=
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2003 Prentice-Hall, Inc.Chap 10-55
Prediction of Individual Values
Prediction Interval for IndividualResponse Yi at a Particular Xi
Addition of 1 increases width of intervalfrom that for the mean of Y
2
2
2
1
( )1 1
( )
ii n YX n
i
i
X XY t Sn
X X
=
+ +
Inte l E tim te fo Diffe ent
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2003 Prentice-Hall, Inc.Chap 10-56
Interval Estimates for DifferentValues ofX
Y
X
Prediction Interval
for a individual Yi
A givenX
Confidence
Interval for the
mean ofY
Yi=b0+
b1Xi
X
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2003 Prentice-Hall, Inc.Chap 10-57
Example: Produce Stores
Data for 7 Stores:
Regression Equation
Obtained:
Consider a storewith 2000 squarefeet.
AnnualStore 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
1636.415 1.487i
Y X= +
Estimation of Mean Values:
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2003 Prentice-Hall, Inc.Chap 10-58
Estimation of Mean Values:Example
Find the 95% confidence interval for the average annualsales for stores of 2,000 square feet
2
22
1
( )1 4610.45 612.66
( )
ii n YX n
i
i
X XY t S
nX X
=
+ =
Predicted Sales
Confidence Interval Estimate for | iY X X =
( ) 1636.415 1.487 4610.45 $000iY X= + =
2 52350.29 611.75 2.5706YX nX S t t= = = =
Prediction Interval for Y :
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2003 Prentice-Hall, Inc.Chap 10-59
Prediction Interval for Y:Example
Find the 95% prediction interval for annual sales of oneparticular store of 2,000 square feet
Predicted Sales)
2
2
2
1
( )1 1 4610.45 1687.68
( )
ii n YX n
i
i
X XY t S
nX X
=
+ + =
Prediction Interval for IndividualiX X
Y =
( ) 1636.415 1.487 4610.45 $000iY X= + =
2 52350.29 611.75 2.5706YX nX S t t= = = =
Estimation of Mean Values and Prediction
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2003 Prentice-Hall, Inc.Chap 10-60
of Individual Values in PHStat
In Excel, use PHStat | Regression | Simple
Linear Regression
Check the Confidence and Prediction Interval forX= box
EXCEL Spreadsheet of Regression Sales on
Footage
Microsoft Excel
Worksheet
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2003 Prentice-Hall, Inc.Chap 10-61
Pitfalls of Regression Analysis
Lacking an Awareness of the AssumptionsUnderlining Least-squares Regression
Not Knowing How to Evaluate the Assumptions Not Knowing What the Alternatives to Least-
squares Regression are if a ParticularAssumption is Violated
Using a Regression Model Without Knowledgeof the Subject Matter
Strategy for Avoiding the Pitfalls
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2003 Prentice-Hall, Inc.Chap 10-62
Strategy for Avoiding the Pitfallsof Regression
Start with a scatter plot of X on Y to observepossible relationship
Perform residual analysis to check theassumptions
Use a histogram, stem-and-leaf display, box-
and-whisker plot, or normal probability plot ofthe residuals to uncover possible non-normality
Strategy for Avoiding the Pitfalls
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Strategy for Avoiding the Pitfallsof Regression
If there is violation of any assumption, usealternative methods (e.g., least absolutedeviation regression or least median of squares
regression) to least-squares regression oralternative least-squares models (e.g.,curvilinear or multiple regression)
If there is no evidence of assumption violation,then test for the significance of the regressioncoefficients and construct confidence intervalsand prediction intervals
(continued)
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Chapter Summary
Introduced Types of Regression Models
Discussed Determining the Simple LinearRegression Equation
Described Measures of Variation
Addressed Assumptions of Regression andCorrelation
Discussed Residual Analysis
Addressed Measuring Autocorrelation
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Chapter Summary
Described Inference about the Slope
Discussed Correlation - Measuring theStrength of the Association
Addressed Estimation of Mean Values andPrediction of Individual Values
Discussed Pitfalls in Regression and Ethical
Issues
(continued)
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