Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 12
Simple Linear Regression
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-1
Correction from last week:
ANOVA=t-test
Source of
variation
Between
(2 groups)
Within
d.f.
1
2n-2
Sum of
squares
SSA
Squared
(squared difference
difference in means
in means
multiplied
by n)
SSW
equivalent to
numerator of
pooled
variance
Total
2n-1
variation
Mean
Sum of
Squares
TSS
Pooled
variance
F-statistic
n( X  Y ) 2
sp
2
(
Go to
(X  Y )
sp
n
2

sp
n
p-value
)  (t 2 n  2 )
2
2
2
F1, 2n-2
Chart
notice
values
are just (t
2
2n-2)
n
X n  Yn 2
X Y
SSB  n (X n  (
))  n (Yn  ( n n )) 2 
2
2
i 1
i 1
n


n
n
X n Yn 2
Y X
n (
 )  n ( n  n )2
2
2
2
i 1
i 1 2

SSB algebra

X n 2 Yn 2
X *Y
Y
X
X *Y
)  ( )  2 n n  ( n )2  ( n )2  2 n n ) 
2
2
2
2
2
2
2
2
n( X n  X n * Yn  Yn )  n( X n  Yn ) 2
n((
Source of
variation
Between
(2 groups)
Within
d.f.
1
2n-2
Sum of
squares
SSB
Squared
(squared difference
difference in means
in means
times n
multiplied
by n)
SSW
equivalent to
numerator of
pooled
variance
Total
2n-1
variation
Mean
Sum of
Squares
TSS
Pooled
variance
F-statistic
p-value
Go to
n( X  Y )
sp
2
2
(
(X  Y )
sp
n
2

sp
n
2
) 2  (t 2 n  2 ) 2
F1, 2n-2
Chart
notice
values
are just (t
2
2n-2)
Chapter Goals
After completing this chapter, you should be
able to:
 Explain the simple linear regression model
 Obtain and interpret the simple linear regression
equation for a set of data
 Evaluate regression residuals for aptness of the fitted
model
 Understand the assumptions behind regression
analysis
 Explain measures of variation and determine whether
the independent variable is significant
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-4
Chapter Goals
(continued)
After completing this chapter, you should be
able to:
 Calculate and interpret confidence intervals for the
regression coefficients
 Use the Durbin-Watson statistic to check for
autocorrelation
 Form confidence and prediction intervals around an
estimated Y value for a given X
 Recognize some potential problems if regression
analysis is used incorrectly
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-5
Correlation vs. Regression
 A scatter plot (or scatter diagram) can be used
to show the relationship between two variables
 Correlation analysis is used to measure
strength of the association (linear relationship)
between two variables
 Correlation is only concerned with strength of the
relationship
 No causal effect is implied with correlation
 Correlation was first presented in Chapter 3
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-6
Introduction to
Regression Analysis
 Regression analysis is used to:
 Predict the value of a dependent variable based on the
value of at least one independent variable
 Explain the impact of changes in an independent
variable on the dependent variable
Dependent variable: the variable we wish to explain
Independent variable: the variable used to explain
the dependent variable
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-7
Simple Linear Regression
Model
 Only one independent variable, X
 Relationship between X and Y is
described by a linear function
 Changes in Y are assumed to be caused
by changes in X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-8
Types of Relationships
Linear relationships
Y
Curvilinear relationships
Y
X
Y
X
Y
X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
X
Chap 12-9
Types of Relationships
(continued)
Strong relationships
Y
Weak relationships
Y
X
Y
X
Y
X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
X
Chap 12-10
Types of Relationships
(continued)
No relationship
Y
X
Y
X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-11
Simple Linear Regression
Model
The population regression model:
Population
Y intercept
Dependent
Variable
Population
Slope
Coefficient
Independent
Variable
Random
Error
term
Yi  β0  β1Xi  ε i
Linear component
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Random Error
component
Chap 12-12
Simple Linear Regression
Model
(continued)
Y
Yi  β0  β1Xi  ε i
Observed Value
of Y for Xi
εi
Predicted Value
of Y for Xi
Slope = β1
Random Error
for this Xi value
Intercept = β0
Xi
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
X
Chap 12-13
Simple Linear Regression
Equation
The simple linear regression equation provides an
estimate of the population regression line
Estimated
(or predicted)
Y value for
observation i
Estimate of
the regression
intercept
Estimate of the
regression slope
Ŷi  b0  b1Xi
Value of X for
observation i
The individual random error terms ei have a mean of zero
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-14
Least Squares Method
 b0 and b1 are obtained by finding the values
of b0 and b1 that minimize the sum of the
squared differences between Y and Ŷ :
min  (Yi Ŷi )  min  (Yi  (b0  b1Xi ))
2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
2
Chap 12-15
Finding the Least Squares
Equation
 The coefficients b0 and b1 , and other
regression results in this chapter, will be
found using Excel
Formulas are shown in the text at the end of
the chapter for those who are interested
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-16
Interpretation of the
Slope and the Intercept
 b0 is the estimated average value of Y
when the value of X is zero
 b1 is the estimated change in the
average value of Y as a result of a
one-unit change in X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-17
Simple Linear Regression
Example
 A real estate agent wishes to examine the
relationship between the selling price of a home
and its size (measured in square feet)
 A random sample of 10 houses is selected
 Dependent variable (Y) = house price in $1000s
 Independent variable (X) = square feet
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-18
Sample Data for House Price
Model
House Price in $1000s
(Y)
Square Feet
(X)
245
1400
312
1600
279
1700
308
1875
199
1100
219
1550
405
2350
324
2450
319
1425
255
1700
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-19
Graphical Presentation
 House price model: scatter plot
House Price ($1000s)
450
400
350
300
250
200
150
100
50
0
0
500
1000
1500
2000
2500
3000
Square Feet
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-20
Regression Using Excel
 Tools / Data Analysis / Regression
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-21
Excel Output
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
The regression equation is:
house price  98.24833  0.10977 (square feet)
41.33032
Observations
10
ANOVA
df
SS
MS
F
11.0848
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
Standard Error
t Stat
P-value
Significance F
0.01039
Lower 95%
Upper 95%
98.24833
58.03348
1.69296
0.12892
-35.57720
232.07386
0.10977
0.03297
3.32938
0.01039
0.03374
0.18580
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-22
Graphical Presentation
House Price ($1000s)
 House price model: scatter plot and
regression
line
450
Intercept
= 98.248
400
350
Slope
= 0.10977
300
250
200
150
100
50
0
0
500
1000
1500
2000
2500
3000
Square Feet
house price  98.24833  0.10977 (square feet)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-23
Interpretation of the
Intercept, b0
house price  98.24833  0.10977 (square feet)
 b0 is the estimated average value of Y when the
value of X is zero (if X = 0 is in the range of
observed X values)
 Here, no houses had 0 square feet, so b0 = 98.24833
just indicates that, for houses within the range of
sizes observed, $98,248.33 is the portion of the
house price not explained by square feet
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-24
Interpretation of the
Slope Coefficient, b1
house price  98.24833  0.10977 (square feet)
 b1 measures the estimated change in the
average value of Y as a result of a oneunit change in X
 Here, b1 = .10977 tells us that the average value of a
house increases by .10977($1000) = $109.77, on
average, for each additional one square foot of size
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-25
Predictions using
Regression Analysis
Predict the price for a house
with 2000 square feet:
house price  98.25  0.1098 (sq.ft.)
 98.25  0.1098(200 0)
 317.85
The predicted price for a house with 2000
square feet is 317.85($1,000s) = $317,850
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-26
Interpolation vs. Extrapolation
 When using a regression model for prediction,
only predict within the relevant range of data
Relevant range for
interpolation
House Price ($1000s)
450
400
350
300
250
200
150
100
50
0
0
500
1000
1500
2000
2500
3000
Do not try to
extrapolate
beyond the range
of observed X’s
Square Feet
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-27
Measures of Variation
 Total variation is made up of two parts:
SST  SSR  SSE
Total Sum of
Squares
Regression Sum
of Squares
SST   ( Yi  Y )2
SSR   ( Ŷi  Y )2
Error Sum of
Squares
SSE   ( Yi  Ŷi )2
where:
Y = Average value of the dependent variable
Yi = Observed values of the dependent variable
Ŷi = Predicted value of Y for the given Xi value
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-28
Measures of Variation
(continued)
 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
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-29
Measures of Variation
(continued)
Y
Yi
SSE =
(Yi - Yi )2
Y
_
SST =
(Yi - Y)2
_
Y
SSR =
_
Y
Xi
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
(Yi - Y)2
_
Y
X
Chap 12-30
Coefficient of Determination, r2
 The coefficient of determination is the portion
of the total variation in the dependent variable
that is explained by variation in the
independent variable
 The coefficient of determination is also called
r-squared and is denoted as r2
SSR regression sum of squares
r 

SST
total sum of squares
2
note:
0 r 1
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
2
Chap 12-31
Examples of Approximate
r2 Values
Y
r2 = 1
r2 = 1
X
100% of the variation in Y is
explained by variation in X
Y
r2
=1
Perfect linear relationship
between X and Y:
X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-32
Examples of Approximate
r2 Values
Y
0 < r2 < 1
X
Weaker linear relationships
between X and Y:
Some but not all of the
variation in Y is explained
by variation in X
Y
X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-33
Examples of Approximate
r2 Values
r2 = 0
Y
No linear relationship
between X and Y:
r2 = 0
X
The value of Y does not
depend on X. (None of the
variation in Y is explained
by variation in X)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-34
Excel Output
SSR 18934.9348
r 

 0.58082
SST 32600.5000
2
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
58.08% of the variation in
house prices is explained by
variation in square feet
41.33032
Observations
10
ANOVA
df
SS
MS
F
11.0848
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
Standard Error
t Stat
P-value
Significance F
0.01039
Lower 95%
Upper 95%
98.24833
58.03348
1.69296
0.12892
-35.57720
232.07386
0.10977
0.03297
3.32938
0.01039
0.03374
0.18580
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-35
Standard Error of Estimate
 The standard deviation of the variation of
observations around the regression line is
estimated by
n
S YX
SSE


n2
2
(
Y

Ŷ
)
 i i
i1
n2
Where
SSE = error sum of squares
n = sample size
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-36
Excel Output
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
SYX  41.33032
41.33032
Observations
10
ANOVA
df
SS
MS
F
11.0848
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
Standard Error
t Stat
P-value
Significance F
0.01039
Lower 95%
Upper 95%
98.24833
58.03348
1.69296
0.12892
-35.57720
232.07386
0.10977
0.03297
3.32938
0.01039
0.03374
0.18580
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-37
Comparing Standard Errors
SYX is a measure of the variation of observed
Y values from the regression line
Y
Y
small sYX
X
large sYX
X
The magnitude of SYX should always be judged relative to the
size of the Y values in the sample data
i.e., SYX = $41.33K is moderately small relative to house prices in
the $200 - $300K range
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-38
Assumptions of Regression
 Normality of Error
 Error values (ε) are normally distributed for any given
value of X
 Homoscedasticity
 The probability distribution of the errors has constant
variance
 Independence of Errors
 Error values are statistically independent
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-39
Residual Analysis
ei  Yi  Ŷi
 The residual for observation i, ei, is the difference
between its observed and predicted value
 Check the assumptions of regression by examining the
residuals
 Examine for linearity assumption
 Examine for constant variance for all levels of X
(homoscedasticity)
 Evaluate normal distribution assumption
 Evaluate independence assumption
 Graphical Analysis of Residuals
 Can plot residuals vs. X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-40
Residual Analysis for Linearity
Y
Y
x
x
Not Linear
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
residuals
residuals
x
x

Linear
Chap 12-41
Residual Analysis for
Homoscedasticity
Y
Y
x
x
Non-constant variance
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
residuals
residuals
x
x

Constant variance
Chap 12-42
Residual Analysis for
Independence
Not Independent
X
residuals
residuals
X
residuals

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Independent
X
Chap 12-43
Excel Residual Output
RESIDUAL OUTPUT
Residuals
1
251.92316
-6.923162
2
273.87671
38.12329
3
284.85348
-5.853484
4
304.06284
3.937162
5
218.99284
-19.99284
80
60
40
Residuals
Predicted
House Price
House Price Model Residual Plot
20
0
6
268.38832
-49.38832
-20
7
356.20251
48.79749
-40
8
367.17929
-43.17929
-60
9
254.6674
64.33264
10
284.85348
-29.85348
0
1000
2000
3000
Square Feet
Does not appear to violate
any regression assumptions
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-44
Measuring Autocorrelation:
The Durbin-Watson Statistic
 Used when data are collected over time to
detect if autocorrelation is present
 Autocorrelation exists if residuals in one
time period are related to residuals in
another period
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-45
Autocorrelation
 Autocorrelation is correlation of the errors
(residuals) over time
Time (t) Residual Plot
 Here, residuals show
a cyclic pattern, not
random
Residuals
15
10
5
0
-5 0
2
4
6
8
-10
-15
Time (t)
 Violates the regression assumption that
residuals are random and independent
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-46
The Durbin-Watson Statistic
 The Durbin-Watson statistic is used to test for
autocorrelation
H0: residuals are not correlated
H1: autocorrelation is present
n
D
 (e  e
i 2
i
i1
)
2
 The possible range is 0 ≤ D ≤ 4
 D should be close to 2 if H0 is true
n
2
e
 i
i1
 D less than 2 may signal positive
autocorrelation, D greater than 2 may
signal negative autocorrelation
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-47
Testing for Positive
Autocorrelation
H0: positive autocorrelation does not exist
H1: positive autocorrelation is present
 Calculate the Durbin-Watson test statistic = D
(The Durbin-Watson Statistic can be found using PHStat in Excel)
 Find the values dL and dU from the Durbin-Watson table
(for sample size n and number of independent variables k)
Decision rule: reject H0 if D < dL
Reject H0
0
Inconclusive
dL
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Do not reject H0
dU
2
Chap 12-48
Testing for Positive
Autocorrelation
(continued)
160
 Example with n = 25:
140
120
Excel/PHStat output:
Sales
100
Durbin-Watson Calculations
Sum of Squared
Difference of Residuals
y = 30.65 + 4.7038x
2
R = 0.8976
80
60
3296.18
Sum of Squared
Residuals
40
20
3279.98
0
0
Durbin-Watson
Statistic
5
10
15
20
25
30
Tim e
1.00494
n
D
 (e  e
i2
i
n
 ei
2
i1
)2
3296.18

 1.00494
3279.98
i1
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-49
Testing for Positive
Autocorrelation
(continued)
 Here, n = 25 and there is k = 1 one independent variable
 Using the Durbin-Watson table, dL = 1.29 and dU = 1.45
 D = 1.00494 < dL = 1.29, so reject H0 and conclude that
significant positive autocorrelation exists
 Therefore the linear model is not the appropriate model
to forecast sales
Decision: reject H0 since
D = 1.00494 < dL
Reject H0
0
Inconclusive
dL=1.29
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Do not reject H0
dU=1.45
2
Chap 12-50
Inferences About the Slope
 The standard error of the regression slope
coefficient (b1) is estimated by
S YX
Sb1 

SSX
S YX
2
(X

X
)
 i
where:
Sb1
= Estimate of the standard error of the least squares slope
SSE = Standard error of the estimate
S YX 
n2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-51
Excel Output
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
Sb1  0.03297
41.33032
Observations
10
ANOVA
df
SS
MS
F
11.0848
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
Standard Error
t Stat
P-value
Significance F
0.01039
Lower 95%
Upper 95%
98.24833
58.03348
1.69296
0.12892
-35.57720
232.07386
0.10977
0.03297
3.32938
0.01039
0.03374
0.18580
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-52
Comparing Standard Errors of
the Slope
Sb1 is a measure of the variation in the slope of regression
lines from different possible samples
Y
Y
small Sb1
X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
large Sb1
X
Chap 12-53
Inference about the Slope:
t Test
 t test for a population slope
 Is there a linear relationship between X and Y?
 Null and alternative hypotheses
H0: β1 = 0
H1: β1  0
(no linear relationship)
(linear relationship does exist)
 Test statistic
b1  β1
t
Sb1
d.f.  n  2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
where:
b1 = regression slope
coefficient
β1 = hypothesized slope
Sb1 = standard
error of the slope
Chap 12-54
Inference about the Slope:
t Test
(continued)
House Price
in $1000s
(y)
Square Feet
(x)
245
1400
312
1600
279
1700
308
1875
199
1100
219
1550
405
2350
324
2450
319
1425
255
1700
Estimated Regression Equation:
house price  98.25  0.1098 (sq.ft.)
The slope of this model is 0.1098
Does square footage of the house
affect its sales price?
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-55
Inferences about the Slope:
t Test Example
H0: β1 = 0
H1: β1  0
From Excel output:
Coefficients
Intercept
Square Feet
b1
Standard Error
Sb1
t Stat
P-value
98.24833
58.03348
1.69296
0.12892
0.10977
0.03297
3.32938
0.01039
b1  β1 0.10977  0
t

 3.32938
t
Sb1
0.03297
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-56
Inferences about the Slope:
t Test Example
(continued)
Test Statistic: t = 3.329
H0: β1 = 0
H1: β1  0
From Excel output:
Coefficients
Intercept
Square Feet
b1
Standard Error
Sb1
t
t Stat
P-value
98.24833
58.03348
1.69296
0.12892
0.10977
0.03297
3.32938
0.01039
d.f. = 10-2 = 8
a/2=.025
Reject H0
a/2=.025
Do not reject H0
-tα/2
-2.3060
0
Reject H
0
tα/2
2.3060 3.329
Decision:
Reject H0
Conclusion:
There is sufficient evidence
that square footage affects
house price
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-57
Inferences about the Slope:
t Test Example
(continued)
P-value = 0.01039
H0: β1 = 0
H1: β1  0
From Excel output:
Coefficients
Intercept
Square Feet
This is a two-tail test, so
the p-value is
P(t > 3.329)+P(t < -3.329)
= 0.01039
(for 8 d.f.)
P-value
Standard Error
t Stat
P-value
98.24833
58.03348
1.69296
0.12892
0.10977
0.03297
3.32938
0.01039
Decision: P-value < α so
Reject H0
Conclusion:
There is sufficient evidence
that square footage affects
house price
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-58
F-Test for Significance
 F Test statistic:
where
MSR
F
MSE
MSR 
SSR
k
SSE
MSE 
n  k 1
where F follows an F distribution with k numerator and (n – k - 1)
denominator degrees of freedom
(k = the number of independent variables in the regression model)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-59
Excel Output
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
MSR 18934.9348
F

 11.0848
MSE 1708.1957
41.33032
Observations
10
With 1 and 8 degrees
of freedom
P-value for
the F-Test
ANOVA
df
SS
MS
F
11.0848
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
Standard Error
t Stat
P-value
Significance F
0.01039
Lower 95%
Upper 95%
98.24833
58.03348
1.69296
0.12892
-35.57720
232.07386
0.10977
0.03297
3.32938
0.01039
0.03374
0.18580
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-60
F-Test for Significance
(continued)
Test Statistic:
H0: β1 = 0
H1: β1 ≠ 0
 = .05
df1= 1
df2 = 8
MSR
F
 11.08
MSE
Decision:
Reject H0 at
Critical
Value:
= 0.05
F = 5.32
Conclusion:
= .05
0
Do not
reject H0
Reject H0
F
There is sufficient evidence that
house size affects selling price
F.05 = 5.32
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-61
Confidence Interval Estimate
for the Slope
Confidence Interval Estimate of the Slope:
b1  t n2Sb1
d.f. = n - 2
Excel Printout for House Prices:
Intercept
Square Feet
Coefficients
Standard Error
t Stat
P-value
98.24833
0.10977
Lower 95%
Upper 95%
58.03348
1.69296
0.12892
-35.57720
232.07386
0.03297
3.32938
0.01039
0.03374
0.18580
At 95% level of confidence, the confidence interval for
the slope is (0.0337, 0.1858)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-62
Confidence Interval Estimate
for the Slope
(continued)
Intercept
Square Feet
Coefficients
Standard Error
t Stat
P-value
98.24833
0.10977
Lower 95%
Upper 95%
58.03348
1.69296
0.12892
-35.57720
232.07386
0.03297
3.32938
0.01039
0.03374
0.18580
Since the units of the house price variable is
$1000s, we are 95% confident that the average
impact on sales price is between $33.70 and
$185.80 per square foot of house size
This 95% confidence interval does not include 0.
Conclusion: There is a significant relationship between
house price and square feet at the .05 level of significance
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-63
t Test for a Correlation Coefficient
 Hypotheses
H0: ρ = 0
HA: ρ ≠ 0
(no correlation between X and Y)
(correlation exists)
 Test statistic

t
r -ρ
1 r
n2
2
(with n – 2 degrees of freedom)
where
r   r 2 if b1  0
r   r 2 if b1  0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-64
Example: House Prices
Is there evidence of a linear relationship
between square feet and house price at the
.05 level of significance?
H0: ρ = 0
H1: ρ ≠ 0
(No correlation)
(correlation exists)
=.05 , df = 10 - 2 = 8
t
r ρ
1 r 2
n2

.762  0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
1  .762 2
10  2
 3.33
Chap 12-65
Example: Test Solution
t
r ρ
1 r 2
n2

.762  0
1  .762 2
10  2
 3.33
Conclusion:
There is
evidence of a
linear association
at the 5% level of
significance
d.f. = 10-2 = 8
a/2=.025
Reject H0
-tα/2
-2.3060
a/2=.025
Do not reject H0
0
Reject H0
tα/2
2.3060
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Decision:
Reject H0
3.33
Chap 12-66
Estimating Mean Values and
Predicting Individual Values
Goal: Form intervals around Y to express
uncertainty about the value of Y for a given Xi
Confidence
Interval for
the mean of
Y, given Xi
Y
Y
Y = b0+b1Xi
Prediction Interval
for an individual Y,
given Xi
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Xi
X
Chap 12-67
Confidence Interval for
the Average Y, Given X
Confidence interval estimate for the
mean value of Y given a particular Xi
Confidence interval for μY|X  Xi :
Ŷ  t n2S YX hi
Size of interval varies according
to distance away from mean, X
1 (Xi  X)2 1
(Xi  X)2
hi  
 
n
SSX
n  (Xi  X)2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-68
Prediction Interval for
an Individual Y, Given X
Confidence interval estimate for an
Individual value of Y given a particular Xi
Confidence interval for YX Xi :
Ŷ  t n2S YX 1  hi
This extra term adds to the interval width to reflect
the added uncertainty for an individual case
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-69
Estimation of Mean Values:
Example
Confidence Interval Estimate for μY|X=X
i
Find the 95% confidence interval for the mean price
of 2,000 square-foot houses
Predicted Price Yi = 317.85 ($1,000s)
Ŷ  t n-2S YX
1
(Xi  X)2

 317.85  37.12
2
n  (Xi  X)
The confidence interval endpoints are 280.66 and 354.90,
or from $280,660 to $354,900
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-70
Estimation of Individual Values:
Example
Prediction Interval Estimate for YX=X
i
Find the 95% prediction interval for an individual
house with 2,000 square feet
Predicted Price Yi = 317.85 ($1,000s)
Ŷ  t n-1S YX
1
(Xi  X)2
1 
 317.85  102.28
2
n  (Xi  X)
The prediction interval endpoints are 215.50 and 420.07,
or from $215,500 to $420,070
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-71
Finding Confidence and
Prediction Intervals in Excel
 In Excel, use
PHStat | regression | simple linear regression …
 Check the
“confidence and prediction interval for X=”
box and enter the X-value and confidence level
desired
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-72
Finding Confidence and
Prediction Intervals in Excel
(continued)
Input values
Y
Confidence Interval Estimate for μY|X=Xi
Prediction Interval Estimate for YX=Xi
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-73
Pitfalls of Regression Analysis
 Lacking an awareness of the assumptions
underlying least-squares regression
 Not knowing how to evaluate the assumptions
 Not knowing the alternatives to least-squares
regression if a particular assumption is violated
 Using a regression model without knowledge of
the subject matter
 Extrapolating outside the relevant range
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-74
Strategies for Avoiding
the Pitfalls of Regression
 Start with a scatter plot of X on Y to observe
possible relationship
 Perform residual analysis to check the
assumptions
 Plot the residuals vs. X to check for violations of
assumptions such as homoscedasticity
 Use a histogram, stem-and-leaf display, box-andwhisker plot, or normal probability plot of the
residuals to uncover possible non-normality
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-75
Strategies for Avoiding
the Pitfalls of Regression
(continued)
 If there is violation of any assumption, use
alternative methods or models
 If there is no evidence of assumption violation,
then test for the significance of the regression
coefficients and construct confidence intervals
and prediction intervals
 Avoid making predictions or forecasts outside
the relevant range
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-76
Chapter Summary
 Introduced types of regression models
 Reviewed assumptions of regression and
correlation
 Discussed determining the simple linear
regression equation
 Described measures of variation
 Discussed residual analysis
 Addressed measuring autocorrelation
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-77
Chapter Summary
(continued)
 Described inference about the slope
 Discussed correlation -- measuring the strength
of the association
 Addressed estimation of mean values and
prediction of individual values
 Discussed possible pitfalls in regression and
recommended strategies to avoid them
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-78
Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 13
Introduction to Multiple Regression
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-79
Chapter Goals
After completing this chapter, you should be
able to:
 analyze and interpret the computer output for a
multiple regression model
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-80
The Multiple Regression
Model
Idea: Examine the linear relationship between
1 dependent (Y) & 2 or more independent variables (Xi)
Multiple Regression Model with k Independent Variables:
Y-intercept
Population slopes
Random Error
Yi  β0  β1X1i  β2 X2i    βk Xki  ε
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-81
Multiple Regression Equation
The coefficients of the multiple regression model are
estimated using sample data
Multiple regression equation with k independent variables:
Estimated
(or predicted)
value of Y
Estimated
intercept
Estimated slope coefficients
Ŷi  b0  b1X1i  b2 X2i    bk Xki
In this chapter we will always use Excel to obtain the
regression slope coefficients and other regression
summary measures.
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-82
Multiple Regression Equation
(continued)
Two variable model
Y
Ŷ  b0  b1X1  b2 X2
X2
X1
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-83
Example:
2 Independent Variables
 A distributor of frozen desert pies wants to
evaluate factors thought to influence demand
 Dependent variable:
Pie sales (units per week)
 Independent variables: Price (in $)
Advertising ($100’s)
 Data are collected for 15 weeks
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-84
Pie Sales Example
Week
Pie
Sales
Price
($)
Advertising
($100s)
1
350
5.50
3.3
2
460
7.50
3.3
3
350
8.00
3.0
4
430
8.00
4.5
5
350
6.80
3.0
6
380
7.50
4.0
7
430
4.50
3.0
8
470
6.40
3.7
9
450
7.00
3.5
10
490
5.00
4.0
11
340
7.20
3.5
12
300
7.90
3.2
13
440
5.90
4.0
14
450
5.00
3.5
15
300
7.00
2.7
Multiple regression equation:
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Sales = b0 + b1 (Price)
+ b2 (Advertising)
Chap 12-85
Estimating a Multiple Linear
Regression Equation
 Excel will be used to generate the coefficients
and measures of goodness of fit for multiple
regression
 Excel:
 Tools / Data Analysis... / Regression
 PHStat:
 PHStat / Regression / Multiple Regression…
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-86
Multiple Regression Output
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
ANOVA
Regression
Sales  306.526 - 24.975(Pri ce)  74.131(Advertising)
15
df
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
Advertising
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 12-87
The Multiple Regression Equation
Sales  306.526 - 24.975(Pri ce)  74.131(Advertising)
where
Sales is in number of pies per week
Price is in $
Advertising is in $100’s.
b1 = -24.975: sales
will decrease, on
average, by 24.975
pies per week for
each $1 increase in
selling price, net of
the effects of changes
due to advertising
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
b2 = 74.131: sales will
increase, on average,
by 74.131 pies per
week for each $100
increase in
advertising, net of the
effects of changes
due to price
Chap 12-88
Using The Equation to Make
Predictions
Predict sales for a week in which the selling
price is $5.50 and advertising is $350:
Sales  306.526 - 24.975(Pri ce)  74.131(Adv ertising)
 306.526 - 24.975 (5.50)  74.131 (3.5)
 428.62
Predicted sales
is 428.62 pies
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Note that Advertising is
in $100’s, so $350
means that X2 = 3.5
Chap 12-89
Predictions in PHStat
 PHStat | regression | multiple regression …
Check the
“confidence and
prediction interval
estimates” box
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-90
Predictions in PHStat
(continued)
Input values
<
Predicted Y value
<
Confidence interval for the
mean Y value, given
these X’s
<
Prediction interval for an
individual Y value, given
these X’s
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-91
Coefficient of
Multiple Determination
 Reports the proportion of total variation in Y
explained by all X variables taken together
2
Y.12..k
r
SSR regression sum of squares


SST
total sum of squares
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-92
Multiple Coefficient of
Determination
(continued)
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
Regression
r
15
df
SSR 29460.0


 .52148
SST 56493.3
52.1% of the variation in pie sales
is explained by the variation in
price and advertising
47.46341
Observations
ANOVA
2
Y.12
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
Advertising
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 12-93
Adjusted r2
 r2 never decreases when a new X variable is
added to the model
 This can be a disadvantage when comparing
models
 What is the net effect of adding a new variable?
 We lose a degree of freedom when a new X
variable is added
 Did the new X variable add enough
explanatory power to offset the loss of one
degree of freedom?
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-94
Adjusted r2
(continued)
 Shows the proportion of variation in Y explained by all X
variables adjusted for the number of X variables used

 n  1 
2
r  1  (1  rY.12..k )

 1
 n  k variables)
(where n = sample size,
 k = number of independent
2
adj
 Penalize excessive use of unimportant independent variables
 Smaller than r2
 Useful in comparing among models
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 12-95
Adjusted r2
(continued)
Regression Statistics
Multiple R
0.72213
R Square
0.52148
Adjusted R Square
0.44172
Standard Error
47.46341
Observations
ANOVA
Regression
15
df
2
adj
r
 .44172
44.2% of the variation in pie sales is
explained by the variation in price and
advertising, taking into account the sample
size and number of independent variables
SS
MS
F
2
29460.027
14730.013
Residual
12
27033.306
2252.776
Total
14
56493.333
Coefficients
Standard Error
Intercept
306.52619
114.25389
2.68285
0.01993
57.58835
555.46404
Price
-24.97509
10.83213
-2.30565
0.03979
-48.57626
-1.37392
74.13096
25.96732
2.85478
0.01449
17.55303
130.70888
Advertising
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
t Stat
6.53861
Significance F
P-value
0.01201
Lower 95%
Upper 95%
Chap 12-96
Residuals in Multiple Regression
Two variable model
Sample
observation
Ŷ  b0  b1X1  b2 X2
<
Residual =
ei = (Yi – Yi)
Y
Yi
<
Yi
x2i
X1
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
<
x1i
X2
The best fit equation, Y ,
is found by minimizing the
sum of squared errors, e2
Chap 12-97