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Unformatted text preview: Business Statistics
Fifth Edition Ken Black Chapter 14:
E Simple Regression Analysis These notes are not to be reproduced without the written permission of F. 8, Alt 14.1 Introduction to Simple Regression Analysis [Black, page 544] Obiective:
Model the relationship between a response or dependent variable (y) and
one (or more) predictor or independent variables (x1, x2, . . . , xk). Example: For consumer purchase decisions, let y = market share and x = consumer’s degree of top of mind brand awareness
(% of consumers who name this brand ﬁrst). Example: For a particular corporation, let
y = sales revenue for a region at the end of the year and
x = advertising expenditure throughout the year for that region. y is recorded in tens of thousands of dollars and x is recorded in thousands of dollars. In Chapter 15, we will use another predictor (average family
disposable income for each region). Refer to the data below. Region Sales Adv Exp A 1 1
B 1 2
C 2 1
D 2 3
E 3 2
F 3 4
G 4 3
H 4 5
I 5 5
J 5 6 These notes are not to be reproduced without the written permission of F. B. Alt o The Appropriateness of 3 Linear Model [Blaclg page 544]
0 Look at a scatterplot. Example (Sales and Advertising Eexpenditures): a. Plot the (x, y) data. Scatterplot for Sale vs. Advertising Expenditures Is a linear model appropriate? 0 Even if a linear model is approE]; all the points (do do noti fall on the
ﬁtted line because we have a 0 An example ofa E] Total Costs = Fixed Costs + Variable Costs. These notes are not to be reproduced without the written permission of F. B. Alt 3 is: 14.2 Determining the Equation of the Regression Line [Black, page 545] a o The general expression for a ﬁtted line is: )7 = 170 + 51x . [In 2nd grade, you had y = ] o How do you ﬁt the line through the points in the scatterplot? o Residuals o Residuals are prediction errors in the sample. 0 The residual for observation (xi, y) is deEP as follows:
= yl. ~ (be + blxl.)
RESIDUAL,
= y,. — 3», o For an arbitrarily ﬁtted line passing through (27,? ), note the residuals
in the scatterplot on page 3. o Criterion for ﬁtting the line through the points in the scatterplot
o Minimize the sum of the squared residuals or min 20’. ‘54)2 E] o Criterion used to obtain [)0 arid bl : These notes are not to be reproduced without the written permission of F. 8. Alt 4 0 b0 arid b1 can be found by solving the following two expressions [Black, page 547]: b1=2(x,~ — no», — W we * and These notes are not to be reproduced without the written permission of F. B. Alt 5 Examgle (Sales and Advertising Eexpenditures): b. Find the leastsquares regression line. Do the calculations by hand. Region SalesgyzAdv Exptxz mm 3c: A 1 1 1 1
B 1 2 2 4
C 2 1 2 1
D 2 3 6 9
E 3 2 6 4
F 3 4 12 16
G 4 3 12 9
H 4 5 20 25
I 5 5 25 25
J 5 6 30 36
Sum E bi =E(xi ~ Y)(y.~  7)/ 309  5‘7)?‘ Note: E] 2(x, ~ m. — y) = 2m we and E
. . *2 The equE} of the least—squares ﬁtted line is: Note: 2(xi —i)2 is sometimes denoted by SSxx These notes are not to be reproduced without the written permission of F. B. Alt Examgle (Sales and advertising expenditures): 0. Find the leastsquares regression line by using Minitab. Regression Analysis: Sales versus Adv Exp The regression equation is
Sales = 0.681 + 0.725 Adv Exp Predictor Coef SE Coef T P
Constant 0.6812 0.5694 1.20 0.266
Adv Exp 0.7246 0.1579 4.59 0.002
S = 0.829702 R—Sq = 72.5% RSq(adj) = 69.0% d. Use Minitab to plot the leastsquares regression line on the scatterplot. Fitted Regression Line for Sales vs. Adv Exp Sales 2 0.6812 + 0.7246 Adv Exp These notes are not to be reproduced without the written permission of F. B. Alt e. Interpret the slope of the line. @ f. Predict sales for a region that has advertising expenditures of 3 units. @ g. Determine the residual for Region D and illustrate it in the ﬁtted line plot.
Residual = (Actual Sales) — (Predicted Sales)
(2.0) — [0.6812 +(0.725)(3.0)] 2 E — 2.855 Fitted Regression Line for Sales vs. Adv Exp
Sales = 0.6812 + 0.7246 Adv Exp These notes are not to be reproduced without the written permission of F. B. Alt 8 o The ﬁtted values and residuals foEFegions follow. Region Sales AdvExp
A 1 1
B 1 2
C 2 1
D 2 3
E 3 2
F 3 4
G 4 3
H 4 5
I 5 5
J 5 6 E] o How many constraints are there on residuals? ; i o The residuals have FITS
1 .405797101
2130434783
1 .405797101
2855072464
2130434783
3579710145
2855072464
4304347826 RES 0.4057971
1 .13043478
0.594202899
0.85507246
0.869565217
0.57971 014
1 .144927536
0.30434783 4304347826 95652174
5.028985507 2898551 0 The residuals have constraints. and (RES)(AdvExp) 0.405797101
2.260869565
0.594202899
2.56521 7391
1 .739130435
2.31884058
33434782609
1 .521 73913
3.47826087
0.173913043 degrees of freedom. These notes are not to be reproduced without the written permission of EB. Alt 14.5 Coefﬁcient of Determination [Black, page 562] E] 0 Based on explained and unexplained deviation E y,. —)7=()>,—)7)+(y,. n)
E] 0 Notation: (r2 or R2) I With no information on x, use to predict y. Examgle (Sales and Advertising Expenditures): The ﬁtted line plot follows: Fitted Regrasion Line for Sales vs. Adv Exp
Saies = 0.6812 + 0.724 Square both sides; Sum z<yi—;>2[=Ez][;i—;)Z + xii—1f SSTotal = SSRegression + SS(Residual)Error These notes are not to be reproduced without the written permission of F. B. Alt 10 o R2 (E Coefﬁcient of Determination) 2
Z i g ._ “SST— 2
Z yi"y Examgle (Sales and Advertising Expenditures):
k. Find the value of the coefﬁcient of determination and interpret it. R Regression Analysis: Sales versus Adv Exp The regression equation is
[ides = 0.681 + 0.725 Adv Exp 8 = 0.829702 RSq = 72.5% R—Sq(adj) = 69.0% Analysis of Variance Source DF SS MS F P
Regression 1 14.493 14.493 21.05 0.002
Residual Error 8 5.507 0.688 Total 9 20.000 Value: [E]
Inte retation: E o r scorrelation coefﬁcient 0 r = (+, ) \i R2 : use the sign of the slope coefﬁcient Examgle (Sales and Advertising Eexpenditures): FE] These notes are not to be reproduced without the written permission of F. B. Alt 11 0 Procedure for using Minitab to do Scatterplot Graph > Scatterplot > Simple
y variable (enter column where y values are)
x variable ( enter column where x values are) 0 Procedure for using Minitab to do Regression Analysis Stat > Regression > Regression Response [enter column where y values are] Predictors [enter column where x values are]
Storage 0 Check off Residuals if so desired
0 Check off Fits if so desired. 0 Procedure for using Minitab to obtain Fitted Line Plot Stat > Regression > Fitted Line Plot
Response [enter column where y values are]
Predictors [enter column where x values are] These notes are not to be reproduced without the written permission of F. B. Alt 12 ...
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This note was uploaded on 09/07/2011 for the course BMGT 231 taught by Professor Staff during the Fall '08 term at Maryland.
 Fall '08
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