ken black qa ch15
Post on 06-Apr-2018
267 Views
Preview:
TRANSCRIPT
-
8/3/2019 Ken Black QA ch15
1/24
Business Statistics, 5th ed.
by Ken Black
Chapter 15
Multiple RegressionAnalysis
Discrete Distributions
PowerPoint presentations prepared by Lloyd Jaisingh,Morehead State University
-
8/3/2019 Ken Black QA ch15
2/24
Learning Objectives
Develop a multiple regression model.
Understand and apply significance tests of the regressionmodel and its coefficients.
Compute and interpret residuals, the standard error of theestimate, and the coefficient of determination.
Interpret multiple regression computer output.
-
8/3/2019 Ken Black QA ch15
3/24
Regression Models
Probabilistic Multiple Regression Model
Y = 0 + 1X1+ 2X2+ 3X3 + . . . + kXk+
Y= the value of the dependent (response) variable
0 = the regression constant1 = the partial regression coefficient of independent variable 12 = the partial regression coefficient of independent variable 2k = the partial regression coefficient of independent variable kk = the number of independent variables
= the error of prediction
-
8/3/2019 Ken Black QA ch15
4/24
Estimated Regression Model
st variableindependenofnumber=
tcoefficienregressionofestimate
3tcoefficienregressionofestimate
2tcoefficienregressionofestimate
1tcoefficienregressionofestimate
constantregressionofestimate
ofvaluepredicted:
3
2
1
0
3322110
k
k
YYwhere
Y
b
b
b
b
b
XbXbXbXbb
k
kk
-
8/3/2019 Ken Black QA ch15
5/24
Multiple Regression Model with Two
Independent Variables (First-Order)
Y
where
Y
where Y
X X
b b X b X
b
b
b
0 1 1 2 2
0
1
2
0 1 1 2 2
0
1
2
:
:
= the regression constant
the partial regression coefficient for independent variable 1
the partial regression coefficient for independent variable 2
= the error of prediction
predicted value of Yestimate of regression constant
estimate of regression coefficient 1
estimate of regression coefficient 2
PopulationModel
Estimated
Model
-
8/3/2019 Ken Black QA ch15
6/24
Response Plane for First-Order Two-
Predictor Multiple Regression Model
X1X2
Response Plane
Y1
Vertical InterceptY
-
8/3/2019 Ken Black QA ch15
7/24
Least Squares Equations for k = 2
0 1 1 2 2
0 1 1 1
2
2 1 2 1
0 2 1 1 2 2 2
2
2
b b X b X Y
b X b X b X X X Y
b X b X X b X X Y
n
-
8/3/2019 Ken Black QA ch15
8/24
Real Estate Data
Observation Y X1 X2 Observation Y X1 X21 63.0
65.1
1,605 35 13 79.7 2,121 14
2 2,489 45 14 84.5 2,485 93 69.9
7
1,553 20 15 96.0 2,300 19
4 76.8 2,404 32 16 109.5 2,714 4
5 73.9 1,884 25 17 102.5 2,463 5
6 77.9 1,558 14 18 121.0 3,076 7
7 74.9 1,748 8 19 104.9 3,048 3
8 78.0 3,105 10 20 128.0 3,267 6
9 79.0 1,682 28 21 129.0 3,069 1010 63.4 2,470 30 22 117.9 4,765 11
11 79.5 1,820 2 23 140.0 4,540 8
12 83.9 2,143 6
Market
Price
($1,000)
Square
Feet
Age
(Years)
Market
Price
($1,000)
Square
Feet
Age
(Years)
-
8/3/2019 Ken Black QA ch15
9/24
Real Estate Data
5000
4000
Price
60
90
120
3000
150
SquareFeet0200015
3045Age
Real Estate Data
-
8/3/2019 Ken Black QA ch15
10/24
MINITAB Output
for the Real Estate Example
The regression equation is
Price = 57.4 + 0.0177 Sq.Feet - 0.666 Age
Predictor Coef StDev T P
Constant 57.35 10.01 5.73 0.000
Sq.Feet 0.017718 0.003146 5.63 0.000Age -0.6663 0.2280 -2.92 0.008
S = 11.96 R-Sq = 74.1% R-Sq(adj) = 71.5%
Analysis of Variance
Source DF SS MS F PRegression 2 8189.7 4094.9 28.63 0.000
Residual Error 20 2861.0 143.1
Total 22 11050.7
-
8/3/2019 Ken Black QA ch15
11/24
Predicting the Price of Home
dollarsthousand658.93
12666.025000177.04.57,12and2500
666.00177.04.57
21
21
Y
XXFor
XXY
-
8/3/2019 Ken Black QA ch15
12/24
Evaluating the Multiple
Regression Model
H
H
k
a
01 2 3
0:
:
At least one of the regression coefficients is 0
H
H
H
H
H
H
H
H
a a
a
k
ak
01
1
03
3
02
2
0
0
0
0
0
0
0
0
0
:
:
:
:
:
:
:
:
Significance
Tests for
IndividualRegression
Coefficients
Testing
the
Overall
Model
-
8/3/2019 Ken Black QA ch15
13/24
Testing the Overall Model for the
Real Estate Example
0istscoefficienregressiontheofoneleastAt:
0: 21
0
aH
H
MSRSSR
kMSE
SSE
n kF
MSR
MSE
1
ANOVAdf SS MS F p
Regression 2 8189.723 4094.86 28.63 .000
Residual (Error) 20 2861.017 143.1
Total 22 11050.74
. , ,.
. . ,
01 2 20585
28 63 585
F
FCal
reject H .0
-
8/3/2019 Ken Black QA ch15
14/24
Significance Test
of the Regression
Coefficients forthe Real Estate
Example
H
H
H
H
a
a
01
1
02
2
0
0
0
0
:
:
:
:
tCal = 5.63 > 2.086, reject H0.
Coefficients Std Dev t Stat p
x1 (Sq.Feet) 0.0177 0.003146 5.63 .000
x2 (Age) -0.666 0.2280 -2.92 .008
t.025,20 = 2.086
-
8/3/2019 Ken Black QA ch15
15/24
Residuals and Sum of Squares Error
for the Real Estate Example
SSE
Observation Y Observation Y
1 43.0 42.466 0.534 0.285 13 59.7 65.602 -5.902 34.832
2 45.1 51.465 -6.365 40.517 14 64.5 75.383 -10.883 118.438
3 49.9 51.540 -1.640 2.689 15 76.0 65.442 10.558 111.479
4 56.8 58.622 -1.822 3.319 16 89.5 82.772 6.728 45.265
5 53.9 54.073 -0.173 0.030 17 82.5 77.659 4.841 23.440
6 57.9 55.627 2.273 5.168 18 101.0 87.187 13.813 190.799
7 54.9 62.991 -8.091 65.466 19 84.9 89.356 -4.456 19.858
8 58.0 85.702 -27.702 767.388 20 108.0 91.237 16.763 280.982
9 59.0 48.495 10.505 110.360 21 109.0 85.064 23.936 572.936
10 63.4 61.124 2.276 5.181 22 97.9 114.447 -16.547 273.81511 59.5 68.265 -8.765 76.823 23 120.0 112.460 7.540 56.854
12 63.9 71.322 -7.422 55.092 2861.017
Y Y Y 2
Y Y Y Y Y 2
Y Y
-
8/3/2019 Ken Black QA ch15
16/24
MINITAB Residual Diagnostics for
the Real Estate Problem
Residual
Pe
rcent
30150-15-30
99
90
50
10
1
Fitted Value
Residual
1401201008060
30
15
0
-15
-30
Residual
Frequenc
y
24120-12-24
6.0
4.5
3.0
1.5
0.0
Observation Order
Residual
222018161412108642
30
15
0
-15
-30
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Histogram of the Residuals Residuals Versus the Order o f the Data
Residual Plots for Price
-
8/3/2019 Ken Black QA ch15
17/24
SSE and Standard Error
of the Estimate
eSSSE
n k
where
1
2861
23 2 1
1196.
: n = number of observations
k = number of independent variables
SSE
ANOVA
df SS MS F PRegression 2 8189.7 4094.9 28.63 .000
Residual (Error) 20 2861.0 143.1
Total 22 11050.7
-
8/3/2019 Ken Black QA ch15
18/24
Coefficient of Multiple Determination (R2)
2
2
8189 723
11050 74741
1 1
2861017
11050 74 741
R
R
SSR
SSY
SSE
SSY
.
..
.
. .
SSEANOVA
df SS MS F p
Regression 2 8189.7 4094.89 28.63 .000
Residual (Error) 20 2861.0 143.1
Total 22 11050.7
SSYY SSR
-
8/3/2019 Ken Black QA ch15
19/24
Adjusted R2
adj
SSE
n kSSY
n
R.
.
.. .
21
1
1
1
2861017
23 2 111050 74
23 1
1 285 715
ANOVA
df SS MS F p
Regression 2 8189.7 4094.9 28.63 .000
Residual (Error) 20 2861.0 143.1
Total 22 11050.7
SSYYSSEn-k-1n-1
-
8/3/2019 Ken Black QA ch15
20/24
Demonstration Problem 15.1: Freight
Data
Country
Freight Cargo Shipped by
Road
(Million Short-Ton
Miles)
Length 0f Roads
(Miles)
Number of Commercial
Vehicles
China 278,806 673,239 5,010,000
Brazil 178,359 1,031,693 1,371,127
India 144,000 1,342,000 1,980,000
Germany 138,975 395,367 2,923,000
Italy 125,171 188,597 2,745,500
Spain 105,824 206,271 2,859,438
Mexico 96,049 157,036 3,758,034
-
8/3/2019 Ken Black QA ch15
21/24
Demonstration Problem 15.1: Excel
Output
Regression Statistics
Multiple R 0.812
R Square 0.659
Adjusted R Square 0.488
Standard Error 44273.86677
Observations 7
ANOVA
df SS MS F Sig. F
Regression 2 15148592381 7.57E+09 3.86 0.116
Residual 4 7840701114 1.96E+09
Total 6 22989293495
Coefficients Standard Error t Stat P-value
Intercept -26425.45085 67624.93769 -0.39 0.716
Length 0f Roads 0.101820862 0.043495015 2.34 0.079
Commercial Vehicles 0.04094856 0.017121018 2.39 0.075
-
8/3/2019 Ken Black QA ch15
22/24
Demonstration Problem 15.1: MINITAB
Output
-
8/3/2019 Ken Black QA ch15
23/24
MINITAB Residual Diagnostics for
Demonstration Problem 15.1
Residual
Pe
rcent
100000500000-50000-100000
99
90
50
10
1
Fitted Value
Re
sidual
250000200000150000100000
50000
25000
0
-25000
-50000
Residual
Frequency
50000250000-25000-50000
2.0
1.5
1.0
0.5
0.0
Observation Order
Residual
7654321
50000
25000
0
-25000
-50000
Normal Probability Plot of the Residuals Residuals Versus the Fitted Values
Hist ogram of t he Residuals Residuals Versus t he Order of t he Dat a
Residual Plots for Freight
-
8/3/2019 Ken Black QA ch15
24/24
Copyright 2008 John Wiley & Sons, Inc.All rights reserved. Reproduction or translation
of this work beyond that permitted in section 117of the 1976 United States Copyright Act without
express permission of the copyright owner isunlawful. Request for further information shouldbe addressed to the Permissions Department, JohnWiley & Sons, Inc. The purchaser may makeback-up copies for his/her own use only and notfor distribution or resale. The Publisher assumesno responsibility for errors, omissions, or damagescaused by the use of these programs or from theuse of the information herein.
top related