Linear Regression Example Data
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, 5e © 2008 Prentice-Hall, Inc.
Chap 13-1
Linear Regression Example
Scatterplot
• House price model: scatter plot
House Price ($1000s)
450
400
350
300
250
200
150
100
50
0
0
500
1000
1500
2000
Square Feet
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Chap 13-2
2500
3000
Linear Regression Example
Using Excel
Tools
-------Data Analysis
-------Regression
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Chap 13-3
Linear Regression Example
Excel Output
The regression equation is:
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
house price  98.24833  0.10977 (squarefeet)
41.33032
Observations
10
ANOVA
df
SS
MS
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
Statistics for Managers Using
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Standard Error
t Stat
F
11.0848
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
Chap 13-4
Linear Regression Example
Graphical Representation
• House price model: scatter plot and regression line
House Price ($1000s)
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 (squarefeet)
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Chap 13-5
Linear Regression Example
Interpretation of b0
house price  98.24833  0.10977 (squarefeet)
• b0 is the estimated mean value of Y when
the value of X is zero (if X = 0 is in the range
of observed X values)
• Because the square footage of the house
cannot be 0, the Y intercept has no practical
application.
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Chap 13-6
Linear Regression Example
Interpretation of b1
house price  98.24833  0.10977 (squarefeet)
• b1 measures the mean change in the average value
of Y as a result of a one-unit change in X
• Here, b1 = .10977 tells us that the mean value of a
house increases by .10977($1000) = $109.77, on
average, for each additional one square foot of size
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Chap 13-7
Linear Regression Example
Making Predictions
Predict the price for a house with 2000 square feet:
house price  98.25  0.1098 (sq.ft.)
 98.25  0.1098(2000)
 317.85
The predicted price for a house with 2000 square feet
is 317.85($1,000s) = $317,850
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Chap 13-8
Linear Regression Example
Making Predictions
• 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
Do not try to
extrapolate beyond
the range of
observed X’s
300
250
200
150
100
50
0
Statistics
0 for Managers
500 Using
1000 1500 2000
Microsoft Excel, 5e © 2008
Square Feet
Prentice-Hall, Inc.
2500
3000
Chap 13-9
Measures of Variation
Total variation is made up of two parts:
SST 
Total Sum of
Squares
SSR 
Regression Sum of
Squares
SSE
Error Sum of
Squares
2
ˆ  Y)2 SSE  ( Y  Y
ˆ
SSR   ( Y
)
i
 i i
SST   ( Yi  Y)2
where:
Y
= Mean value of the dependent variable
Yi = Observed values of the dependent variable
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Yˆ i = Predicted value of Y for the given Xi value
Chap 13-10
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 rsquared and is denoted as r2
SSR regressionsum of squares
r 

SST
total sum of squares
2
0  r2  1
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Chap 13-11
Linear Regression Example
Coefficient of Determination, r2
SSR 18934.9348
r 

 0.58082
SST 32600.5000
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
2
58.08% of the variation in house
prices is explained by variation in
square feet
41.33032
Observations
10
ANOVA
df
SS
MS
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
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Standard Error
t Stat
F
11.0848
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
Chap 13-12
Standard Error of Estimate
• The standard deviation of the variation of observations
around the regression line is estimated by
n
SYX
SSE


n2
2
ˆ
(
Y

Y
)
 i i
i 1
Where
SSE = error sum of squares
n = sample size
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Chap 13-13
n2
Linear Regression Example
Standard Error of Estimate
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
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
Coefficients
Intercept
Square Feet
Statistics for Managers Using
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Standard Error
t Stat
F
11.0848
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
Chap 13-14
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
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Chap 13-15
Inferences 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 (no linear relationship)
– H1: β1 ≠ 0 (linear relationship does exist)
where:
• Test statistic
b1  β1
t
Sb1
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d.f. Chapn13-16
2
b1 = regression slope
coefficient
β1 = hypothesized slope
Sb1 = standard
error of the slope
Inferences About the Slope:
t Test Example
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, 5e © 2008
Prentice-Hall, Inc.
Estimated Regression Equation:
house price  98.25  0.1098 (sq.ft.)
The slope of this model is 0.1098
Is there a relationship between the
square footage of the house and its
sales price?
Chap 13-17
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
98.24833
58.03348
1.69296
0.12892
0.10977
0.03297
3.32938
0.01039
b1  β1 0.10977  0
t
t

 3.32938
Sb1
0.03297
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P-value
Chap 13-18
Inferences About the Slope:
t Test Example
Test Statistic: t = 3.329
• H0: β1 = 0
• H1: β1 ≠ 0
d.f. = 10- 2 = 8
a/2=.025
Reject H0
a/2=.025
Do not reject H0
-tα/2
-2.3060
0
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tα/2
2.3060
Reject H0
3.329
Decision: Reject H0
There is sufficient evidence
that square footage affects
house price
Chap 13-19
Inferences About the Slope:
t Test Example
P-Value
From Excel output:
Coefficients
Intercept
• H0: β1 = 0
• H1: β1 ≠ 0
Square Feet
Standard Error
t Stat
98.24833
58.03348
1.69296
0.12892
0.10977
0.03297
3.32938
0.01039
Decision: Reject H0, since p-value < α
There is sufficient evidence that
square footage affects house price.
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P-value
Chap 13-20
F-Test for Significance
•
F Test statistic:
where
MSR
F
MSE
SSR
MSR 
k
SSE
MSE 
n  k 1
where F follows an F distribution with k numerator degrees of freedom
and (n - k - 1) denominator degrees of freedom
(k = the number of independent variables in the regression model)
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Chap 13-21
F-Test for Significance
Excel Output
Regression Statistics
Multiple R
0.76211
R Square
0.58082
Adjusted R Square
0.52842
Standard Error
F
MSR 18934.9348

 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
Regression
1
18934.9348
18934.9348
Residual
8
13665.5652
1708.1957
Total
9
32600.5000
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Chap 13-22
F
11.0848
Significance F
0.01039
F-Test for Significance
•
•
•
•
Test Statistic:
H0: β1 = 0
H1: β1 ≠ 0
a = .05
df1= 1 df2 = 8
MSR
F
 11.08
MSE
Decision:
Reject H0 at a = 0.05
Critical Value:
Fa = 5.32
Conclusion:
a = .05
0
Do not
Reject H0
reject
F Using
= 5.32
0
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Managers
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F
There is sufficient evidence that
house size affects selling price
Chap 13-23
Confidence Interval Estimate
for the Slope
Confidence Interval Estimate of the Slope:
b1  tn2Sb1
d.f. = n - 2
Excel Printout for House Prices:
Coefficients
Intercept
Standard Error
t Stat
P-value
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
Square Feet
At the 95% level of confidence, the confidence interval
for the slope is (0.0337, 0.1858)
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Chap 13-24
Confidence Interval Estimate
for the Slope
Coefficients
Intercept
Square Feet
Standard Error
t Stat
P-value
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
Since the units of the house price variable is $1000s, you
are 95% confident that the mean change in sales price is
between $33.74 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
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Chap 13-25
Estimating Mean Values and Predicting
Individual Values
Confidence
Interval for
the mean of Y,
given Xi
Goal: Form intervals around Y to express
uncertainty about the value of Y for a given Xi
Y

Y

Y = b0+b1Xi
Prediction Interval for an
individual Y, given Xi
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Chap 13-26
Xi
X
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 :
ˆ t S
Y
n2 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
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Chap 13-27
Prediction Interval for
an Individual Y, Given X
Prediction interval estimate for an individual value
of Y given a particular Xi
Predictionintervalfor YX  Xi :
Yˆ  t n  2SYX 1  hi
This extra term adds to the interval width to reflect the
added uncertainty for an individual case
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Chap 13-28
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)
ˆ t S
Y
n- 2 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
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Chap 13-29
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)
ˆ t S
Y
n-1 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
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Chap 13-30
t Test for a Correlation Coefficient
• Hypotheses
– H0: ρ = 0
– H1: ρ ≠ 0
(no correlation between X and Y)
(correlation exists)
• Test statistic
t
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(with n – 2 degrees of freedom)
r -ρ
w here
1 r
n2
2
r   r 2 if b1  0
r   r 2 if b1  0
Chap 13-31
t Test for a Correlation Coefficient
Is there evidence of a linear relationship between
square feet and house price at the .05 level of
significance?
H0: ρ = 0 (No correlation)
H1: ρ ≠ 0 (correlation exists)
a =.05 , df = 10 - 2 = 8
t
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r ρ
1 r2
n2

.762 0
1  .7622
10  2
Chap 13-32
 3.329
t Test for a Correlation Coefficient
Decision:
Reject H0
d.f. = 10- 2 = 8
a/2=.025
Reject H0
a/2=.025
-tα/2
-2.3060
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Do not reject H0
0
Reject H0
tα/2
2.3060
Chap 13-33
3.329
Conclusion:
There is evidence
of a linear
association at the
5% level of
significance
Residual Analysis
ei  Yi  Yˆi
• 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
– Evaluate Independence assumption
– Evaluate Normal distribution assumption
– Examine Equal variance for all levels of X
• Graphical Analysis of Residuals
– Can plot residuals vs. X
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Chap 13-34
Residual Analysis for Linearity
Y
Y
x
x
residuals
residuals
x
Not Linear
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x
Linear
Chap 13-35
Residual Analysis for Independence
X
X
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residuals
residuals
Independent
residuals
Not Independent
Chap 13-36
X
Checking for Normality
• Examine the Stem-and-Leaf Display of the
Residuals
• Examine the Box-and-Whisker Plot of the
Residuals
• Examine the Histogram of the Residuals
• Construct a Normal Probability Plot of the
Residuals
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Chap 13-37
Residual Analysis for
Equal Variance
Y
Y
x
x
residuals
residuals
x
Unequal variance
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x
Equal variance
Chap 13-38
Linear Regression Example
Excel Residual Output
House Price Model Residual Plot
RESIDUAL OUTPUT
Predicted
House Price
80
Residuals
251.92316
-6.923162
60
2
273.87671
38.12329
40
3
284.85348
-5.853484
4
304.06284
3.937162
5
218.99284
-19.99284
-20
6
268.38832
-49.38832
-40
7
356.20251
48.79749
-60
8
367.17929
-43.17929
9
254.6674
64.33264
10
284.85348
-29.85348
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Residuals
1
20
0
0
1000
2000
Square Feet
Does not appear to violate
any regression assumptions
Chap 13-39
3000
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
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Chap 13-40
Autocorrelation
• Autocorrelation is correlation of the errors
(residuals) over time Time (t) Residual Plot

Here, residuals suggest a
cyclic pattern, not
random
Residuals
15
10
5
0
-5 0
-10
2
4
6
8
-15
Time (t)
 Violates the regression assumption that residuals are
statistically independent
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Chap 13-41
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 equal variance
– Use a histogram, stem-and-leaf display, box-andwhisker plot, or normal probability plot of the
residuals to uncover possible non-normality
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Chap 13-42
Strategies for Avoiding
the Pitfalls of Regression
• 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
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Chap 13-43