This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ...
View
Full Document
 Fall '08
 staff
 Least Squares, Regression Analysis, Errors and residuals in statistics, written permission, adv exp, F. B. Alt

Click to edit the document details