complete f o u r t h e d i t i o n business statistics aczel irwin/mcgraw-hill © the mcgraw-hill...
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COMPLETE f o u r t h e d i t i o nBUSINESS STATISTICS
AczelIrwin/McGraw-Hill © The McGraw-Hill Companies, Inc., 1999
10-1
• 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
• Using the Computer
• Summary and Review of Terms
Simple Linear Regression and Correlation10
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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 Statistics
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X
Y
X
Y
X 0
0
0
0
0
Y
X
Y
X
Y
X
Y
Examples of Other Scatterplots
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The inexact nature of the relationship between advertising and sales suggests that a statistical model might be useful in analyzing the relationship.
A statistical model separates the systematic component of a relationship from the random component.
The inexact nature of the relationship between advertising and sales suggests that a statistical model might be useful in analyzing the relationship.
A statistical model separates the systematic component of a relationship from the random component.
Data
Statistical model
Systematic component
+Random
errors
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 regression, 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 regression, the systematic component is the overall linear relationship, and the random component is the variation around the line.
Model Building
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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; and 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; and 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 Model
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The simple linear regression model posits 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 posits 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 Linear Regression Model
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• 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 Regression Model
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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 Squares
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Fitting a Regression Line
X
Y
Data
X
Y
Three errors from a fitted line
X
Y
Three errors from the least squares regression line
e
X
Errors from the least squares regression line are minimized
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.{Error ei Yi Yi
Yi the predicted value of Y for Xi
Y
X
Y b b X 0 1 the fitted regression line
Yi
Yi
Errors in Regression
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Least 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
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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, and Least Squares Estimators
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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 3787719679498 10605 293426944 390185024
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 3787719679498 10605 293426944 390185024
SSx xx
n
SSxy xyx y
n
bSS XY
SS X
b y b x
22
29342694479448
2
2540947552
390185024106605
2
2551402848
151402848
409475521 255333776 1 26
0 1106605
251 255333776)
79448
25
274 85
( )
. .
( .
.
SSx xx
n
SSxy xyx y
n
bSS XY
SS X
b y b x
22
29342694479448
2
2540947552
390185024106605
2
2551402848
151402848
409475521 255333776 1 26
0 1106605
251 255333776)
79448
25
274 85
( )
. .
( .
.
Example 10-1
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5500500045004000350030002500200015001000
M iles
Do
llars
8 000
7 000
6 000
5 000
4 000
3 000
2 000
1 000
R-Squared = 0.965Y = 274.850 + 1.25533X
Regression of Dollars Charged against Miles
MTB > Regress 'Dollars' 1 'Miles';SUBC> Constant.
Regression Analysis
The regression equation isDollars = 275 + 1.26 Miles
Predictor Coef Stdev t-ratio pConstant 274.8 170.3 1.61 0.120Miles 1.25533 0.04972 25.25 0.000
s = 318.2 R-sq = 96.5% R-sq(adj) = 96.4%
Analysis of Variance
SOURCE DF SS MS F pRegression 1 64527736 64527736 637.47 0.000Error 23 2328161 101224Total 24 66855896
MTB > Regress 'Dollars' 1 'Miles';SUBC> Constant.
Regression Analysis
The regression equation isDollars = 275 + 1.26 Miles
Predictor Coef Stdev t-ratio pConstant 274.8 170.3 1.61 0.120Miles 1.25533 0.04972 25.25 0.000
s = 318.2 R-sq = 96.5% R-sq(adj) = 96.4%
Analysis of Variance
SOURCE DF SS MS F pRegression 1 64527736 64527736 637.47 0.000Error 23 2328161 101224Total 24 66855896
Example 10-1: Using the Computer
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The results on the right side are the output created by selecting REGRESSION option from the DATA ANALYSIS toolkit.
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.98243393R Square 0.965176428Adjusted R Square 0.963662359Standard Error 318.1578225Observations 25
ANOVAdf SS MS F Significance F
Regression 1 64527736.8 64527736.8 637.4721586 2.85084E-18Residual 23 2328161.201 101224.4Total 24 66855898
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 274.8496867 170.3368437 1.61356569 0.120259309 -77.51844165 627.217815 -77.51844165 627.217815MILES 1.255333776 0.049719712 25.248211 2.85084E-18 1.152480856 1.358186696 1.152480856 1.358186696
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.98243393R Square 0.965176428Adjusted R Square 0.963662359Standard Error 318.1578225Observations 25
ANOVAdf SS MS F Significance F
Regression 1 64527736.8 64527736.8 637.4721586 2.85084E-18Residual 23 2328161.201 101224.4Total 24 66855898
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 274.8496867 170.3368437 1.61356569 0.120259309 -77.51844165 627.217815 -77.51844165 627.217815MILES 1.255333776 0.049719712 25.248211 2.85084E-18 1.152480856 1.358186696 1.152480856 1.358186696
Example 10-1: Using Computer-Excel
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Residual Analysis. The plot shows the absence of a relationshipbetween the residuals and the X-values (miles).
Residuals vs. Miles
-800
-600
-400
-200
0
200
400
600
0 1000 2000 3000 4000 5000 6000
Miles
Re
sid
ua
ls
Example 10-1: Using Computer-Excel
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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 Variance
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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 Errors of Regression Estimators
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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 Regression
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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 Regression Parameters
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The correlation between two random variables, X and Y, is a measure of the degree of linear association between the two variables.
The population correlation, denoted by, can take on any value from -1 to 1.
The correlation between two random variables, X and Y, is a measure of the degree 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 Correlation
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Y
X
=0
Y
X
=-.8 Y
X
=.8Y
X
=0
Y
X
=-1 Y
X
=1
Illustrations of Correlation
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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: If < 0, b1 < 0 If = 0, b1 = 0 If > 0, b1 >0
Covariance and Correlation
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SUMMARY OUTPUT
Regression StatisticsMultiple R 0.992265946R Square 0.984591707Adjusted R Square 0.98266567Standard Error 0.279761372Observations 10
ANOVAdf SS MS F Significance F
Regression 1 40.0098686 40.0098686 511.2009204 1.55085E-08Residual 8 0.626131402 0.078266425Total 9 40.636
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept -8.762524695 0.594092798 -14.74942084 4.39075E-07 -10.13250603 -7.39254336 -10.13250603 -7.39254336US 1.423636087 0.062965575 22.60975277 1.55085E-08 1.278437117 1.568835058 1.278437117 1.568835058
RESIDUAL OUTPUT
Observation Predicted Y Residuals1 2.057109569 0.2428904312 2.484200395 0.1157996053 3.05365483 -0.153654834 3.480745656 -0.2807456565 3.765472874 -0.0654728746 4.050200091 0.0497999097 4.619654526 0.1803454748 5.758563396 -0.0585633969 7.466926701 -0.466926701
10 8.463471962 0.436528038
X Variable 1 Line Fit Plot
0
5
10
0 5 10 15
X Variable 1
Y
Y
Predicted Y
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.992265946R Square 0.984591707Adjusted R Square 0.98266567Standard Error 0.279761372Observations 10
ANOVAdf SS MS F Significance F
Regression 1 40.0098686 40.0098686 511.2009204 1.55085E-08Residual 8 0.626131402 0.078266425Total 9 40.636
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept -8.762524695 0.594092798 -14.74942084 4.39075E-07 -10.13250603 -7.39254336 -10.13250603 -7.39254336US 1.423636087 0.062965575 22.60975277 1.55085E-08 1.278437117 1.568835058 1.278437117 1.568835058
RESIDUAL OUTPUT
Observation Predicted Y Residuals1 2.057109569 0.2428904312 2.484200395 0.1157996053 3.05365483 -0.153654834 3.480745656 -0.2807456565 3.765472874 -0.0654728746 4.050200091 0.0497999097 4.619654526 0.1803454748 5.758563396 -0.0585633969 7.466926701 -0.466926701
10 8.463471962 0.436528038
X Variable 1 Line Fit Plot
0
5
10
0 5 10 15
X Variable 1
Y
Y
Predicted Y
Example 10-2: Using Computer-Excel
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8 9 10 11 12
2
3
4
5
6
7
8
9
United States
Inte
rna
tiona
l
Y = -8.76252 + 1.42364X R-Sq = 0.9846
Regression Plot
Example 10-2: Regression Plot
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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 theCorrelation Coefficient
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10-27
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
Hypothesis Tests about the Regression Relationship
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10-28
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
Example 10 - 3:
H0 H1
=1.24 - 1
0.21
H0 is not rejected at the 10% level.
We may not conclude that the beta
coefficient is different from 1.
( - )
:
:
( )
.
. .( . , )
1 1
1 1
11
1
114
1 671 114
2
0 05 58
t
b
s b
t
n
Hypothesis Tests for the Regression Slope
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10-29
The coefficient of determination, r2, 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 Deviation
Explained Deviation
Unexplained 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.
10-7 How Good is the Regression?
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10-30
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 Determination
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10-31
10-8 Analysis of Variance and an F 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
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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 changes when x changes.
x or y
x or y
10-9 Residual Analysis and Checking for Model Inadequacies
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• 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 for Prediction
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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 E[Y|X]
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10-35
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 E[Y|X]
Prediction band for E[Y|X]
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Additional Error in Predicting Individual Value of Y
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
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A (1- ) 100% prediction interval for Y:
Example 10 -1 (X = 4000):
(1.2553)(4000)
( )
( )
. . [ , ]
y tn
x xSSX
2
2
2
11
274.85 2.069 1125
4000 3177.9240947557.84
5296 05 676 62 4619.43 5972.67
A (1- ) 100% prediction interval for Y:
Example 10 -1 (X = 4000):
(1.2553)(4000)
( )
( )
. . [ , ]
y tn
x xSSX
2
2
2
11
274.85 2.069 1125
4000 3177.9240947557.84
5296 05 676 62 4619.43 5972.67
Prediction Interval for a Value of Y
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10-38
A (1- ) 100% prediction interval for the E[Y X]:
Example 10 -1 (X = 4000):
(1.2553)(4000)
( )
( )
. . [ , ]
y tn
x xSSX
2
2
2
1
274.85 2.069125
4000 3177.9240947557.84
5296 05 156 48 5139.57 5452.53
A (1- ) 100% prediction interval for the E[Y X]:
Example 10 -1 (X = 4000):
(1.2553)(4000)
( )
( )
. . [ , ]
y tn
x xSSX
2
2
2
1
274.85 2.069125
4000 3177.9240947557.84
5296 05 156 48 5139.57 5452.53
Prediction Interval for the Average Value of Y
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MTB > regress 'Dollars' 1 'Miles' tres in C3 fits in C4;SUBC> predict 4000;SUBC> residuals in C5.Regression Analysis
The regression equation isDollars = 275 + 1.26 Miles
Predictor Coef Stdev t-ratio pConstant 274.8 170.3 1.61 0.120Miles 1.25533 0.04972 25.25 0.000
s = 318.2 R-sq = 96.5% R-sq(adj) = 96.4%
Analysis of Variance
SOURCE DF SS MS F pRegression 1 64527736 64527736 637.47 0.000Error 23 2328161 101224Total 24 66855896
Fit Stdev.Fit 95.0% C.I. 95.0% P.I. 5296.2 75.6 ( 5139.7, 5452.7) ( 4619.5, 5972.8)
MTB > regress 'Dollars' 1 'Miles' tres in C3 fits in C4;SUBC> predict 4000;SUBC> residuals in C5.Regression Analysis
The regression equation isDollars = 275 + 1.26 Miles
Predictor Coef Stdev t-ratio pConstant 274.8 170.3 1.61 0.120Miles 1.25533 0.04972 25.25 0.000
s = 318.2 R-sq = 96.5% R-sq(adj) = 96.4%
Analysis of Variance
SOURCE DF SS MS F pRegression 1 64527736 64527736 637.47 0.000Error 23 2328161 101224Total 24 66855896
Fit Stdev.Fit 95.0% C.I. 95.0% P.I. 5296.2 75.6 ( 5139.7, 5452.7) ( 4619.5, 5972.8)
Using the Computer
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5500500045004000350030002500200015001000
500
0
-500
Miles
Res
ids
700060005000400030002000
500
0
-500
Fits
Res
ids
MTB > PLOT 'Resids' * 'Fits' MTB > PLOT 'Resids' *'Miles'
Plotting on the Computer (1)
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Plotting on the Computer (2)
MTB > HISTOGRAM 'StRes'
210-1-2
8
7
6
5
4
3
2
1
0
StRes
Fre
quen
cy
5500500045004000350030002500200015001000
7000
6000
5000
4000
3000
2000
MilesD
olla
rs
MTB > PLOT 'Dollars' * 'Miles'