Ch 14_Black_A

Ch 14_Black_A - Business Statistics Fifth Edition Ken Black...

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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 least-squares 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 least-squares 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% R-Sq(adj) = 69.0% d. Use Minitab to plot the least-squares 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 R-Sq = 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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Ch 14_Black_A - Business Statistics Fifth Edition Ken Black...

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