Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management

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1 Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 1 Chapters 14, 15 & 16 Professor Ahmad, Ph.D. Department of Management Revsed August 005

2 Chapter 14 Formulas Smple Lnear Regresson Model: y = β0 + β1x + ε Smple Lnear Regresson Equaton: E(y) = β0 + β1x Least Squares Crteron: ( ) Mn y y$ $y = bo + b1x Estmated Smple Lnear Regresson Equaton where $y = the estmated value of the dependent varable b 0 = the y-ntercept and b 1 = the slope of the lne (x x)(y y) b 1 = and b o = y b1x (x x) Sum of Squares Due to Regresson: SSR (x Total Sum of Squares: [ (x x )(y y) ] = x ) SST = (y y) Also: SST = SSR + SSE Sum of Squares Due to Error: y y$ SSE = ( ) Coeffcent of Determnaton: SSR r SSE = Also r = 1 SST SST Sample Correlaton Coeffcent: r = (the sgn of b 1 ) Coeffcent of Determnaton = + where b 1 = the slope of the regresson equaton r Professor Ahmad s Lecture Notes Page

3 t Test for sgnfcance of ndvdual coeffcents n Lnear Regresson H o : β 1 = 0 H a : β1 0 b t - statstc: t = β 1 1 s b 1 where sb 1 (Estmated Standard Devaton of b1) s s sb 1 = and s = MSE Σ(x x) Reject Ho f t < t α or: t > t α (degrees of freedom = n p 1) F Test for Sgnfcance of the Lnear Regresson Model (ANOVA) : β 0 (The model s not sgnfcant) H 1 o = : β 0 (The model s sgnfcant) Ha 1 Source of Sum of Degrees of Mean Test Statstc Varaton Squares Freedom Square F Regresson SSR p MSR Error (Resdual) SSE n - p - 1 MSE MSR MSE Total SST n - 1 Where: p = Number of ndependent varables n = The sample sze Reject Ho f the Test statstc F > Crtcal Fα Confdence Interval Estmate for the Mean Value of y, that s E(y p ) y$ p t s$ p ± α y where Estmated Standard Devaton of ŷ s p s ŷ p 1 (x p x) = s + Remember: s= MSE n Σ(x x) Professor Ahmad s Lecture Notes Page 3

4 Chapter 14 Smple (Bvarate) Lnear Regresson and Correlaton Ahmad, Inc. s a mcrocomputer producer. The followng data represent Ahmad's yearly sales volume and ther advertsng expendture over a perod of 8 years. (Y) (X) Sales Advertsng Year (In $1,000,000) (In $10,000) a. Develop a scatter dagram of sales versus advertsng. b. Use the method of least squares to compute an estmated regresson lne between sales and advertsng. c. If the company's advertsng expendture s $400,000, what s the predcted sales? Gve the answer n dollars. d. What does the slope of the estmated regresson lne ndcate? e. Compute the coeffcent of determnaton and fully nterpret ts meanng. f. Use the F test to determne whether or not the regresson model s sgnfcant. Let α = g. Use the t test to determne whether the slope of the regresson model s sgnfcant. Let α = 0.05 h. Explan the basc assumptons about the error term n regresson.. Develop a 95% confdence nterval for predctng the average sales for the years when $400,000 was spent on advertsng. j. Use Excel and solve the above problems. k. Usng Excel determne the regresson equaton between sales an tme (where 1996 = 1). Professor Ahmad s Lecture Notes Page 4

5 Multple Regresson Model: y = β 0 + β 1 x 1 + β x +... β p x p + ε Multple Regresson Equaton: E(y) = β0 + β1x1 + βx +... βpxp Estmated Regresson Equaton: y$ = b0 + b1x1 + bx bpxp Multple Coeffcent of Determnaton: Chapters 15 and 16 Formulas SSR R = Also SST R = 1 SSE SST Adjusted Multple Coeffcent of Determnaton: R a = 1 - (1 - R n 1 )( n p 1 ) F Statstc for Determnng When to Add or Delete x : SSE( x1) SSE( x1, x) F = 1 SSE( x1, x) n p 1 General F Test for Addng or Deletng Varables: F = SSE( x, x,..., x ) SSE( x, x x, x x ) 1 q 1 q q+ 1 p q SSE( x, x,..., x, x,..., x ) 1 q q+ 1 p n p 1 p H H t Test for sgnfcance of ndvdual coeffcents n Lnear Regresson o a : β = 0 : β 0 t statstc: Decson Rule: For =1,, 3, p b s t = where s b s the estmated Standard Devaton of b b Reject Ho f t < t α or: t > t α (degrees of freedom = n p 1) Usng the p-value approach: Reject Ho f p-value < α Professor Ahmad s Lecture Notes Page 5

6 F Test for Sgnfcance of the Lnear Regresson Model (ANOVA) H o : β = β =... β = 1 p 0 (.e., the regresson model s NOT sgnfcant) H a : At least one of the coeffcents s sgnfcantly dfferent from zero (the regresson model IS sgnfcant) ANOVA Source of Sum of Degrees of Mean Test Statstc Varaton Squares Freedom Square F Regresson SSR p MSR Error (Resdual) SSE n - p - 1 MSE MSR MSE Total SST n - 1 Where: p = Number of ndependent varables n = The sample sze Decson Rule: Reject Ho f the Test statstc F > Crtcal Fα Usng the p-value approach: Reject Ho f p-value < α Professor Ahmad s Lecture Notes Page 6

7 Chapter 15 Problem 1 Introducton to Multple Regresson and Correlaton Ahmad, Inc. s a mcrocomputer producer. The followng data represent Ahmad's yearly sales volume, ther advertsng expendture, and the number of ndvduals n the sales force over a perod of 15 years: (Y) X1 X X3 Sales Advertsng Sales Force Tme Year ($1,000,000) ($10,000) (100) a. Usng Excel, enter the above data n a fle and save the fle. Prnt the fle as well as the results of all of the followng parts. b. Run the correlaton analyss relatng sales (Y) and all of the ndependent varables. (Do not nclude the column of Year.) Explan the results. Dscuss the concept of multcollnearty. c. Run the Regresson analyses relatng sales (Y) and advertsng (X1). Explan the results. d. Run a regresson analyss relatng sales (Y) and two ndependent varables X1 and X. Explan the results. e. Use an F test (α = 0.05) to determne f varable X contrbutes sgnfcantly to the model. (Topc from Chapter Sxteen secton 16.) f. Run a regresson analyss relatng sales (Y) and two ndependent varables X1 and X3. Explan the results. g. Usng the model developed n part "f", predct sales for 004 assumng we are plannng to advertse $700,000. h. Run a regresson analyss relatng sales (Y) and Tme (X3). Explan the results.. Usng the model developed n part "h" predct sales for 008. j. Run a regresson analyss relatng sales (Y) and three ndependent varables X1, X, and X3. Explan the results. Professor Ahmad s Lecture Notes Page 7

8 Problem Interpretaton of Coeffcents and Other Issues n Multple Regresson A multple regresson model relatng the prce of Rawlston, Inc. stock (Y), the number of shares of the company's stocks sold (X 1 n 100s), and the volume of exchange on the New York Stock Exchange (X n mllons) was developed and part of the results are shown below. ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Intercept X X a. Use the output shown above and wrte an equaton that can be used to predct the prce of the stock. b. Interpret the coeffcents of the estmated regresson equaton. c. At 95% confdence, determne whch varables are sgnfcant and whch are not. d. At 95% confdence, test to determne f the regresson model represents a sgnfcant relatonshp between the ndependent varables and the dependent varable. e. If n a gven day, the number of shares of stock that were sold was 94,500 and the volume of exchange on the New York Stock Exchange was 16 mllon, what would you expect the prce of the stock to be? Professor Ahmad s Lecture Notes Page 8

9 Problem 3 Multple Regresson and Qualtatve Independent Varables The followng data s part of a sample taken from the mortalty tables of a lfe nsurance company. Data provde nformaton on how lfe expectancy (dependent varable Y) relates to two ndependent varables: weght (X 1 n pounds) and whether or not the ndvdual s a smoker (X ), where: x 0 = 1 f the ndvdual s a nonsmoker f the ndvdual s a smoker Age Weght Smoker (Y) (X 1 ) (X ) etc. etc. etc. The results of regresson analyss, relatng Y to X 1 and X s shown below. Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 65 ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Intercept Weght Smoker Professor Ahmad s Lecture Notes Page 9

10 a. Use the output shown above and wrte the regresson equaton. b. Interpret the coeffcents of the estmated regresson equaton. c. At 95% confdence, determne whch varables are sgnfcant and whch are not. d. At 95% confdence, test to determne f the regresson model represents a sgnfcant relatonshp between the ndependent varables and the dependent varable. e. Predct the lfe expectancy of a nonsmoker who weghs 150 pounds. f. Predct the lfe expectancy of a person who smokes 1 pack of cgarettes per day and weghs 150 pounds. g. Predct the lfe expectancy of a person who smokes 3 packs of cgarettes per day and weghs 150 pounds. Professor Ahmad s Lecture Notes Page 10

11 Chapter 16 Problem 1 Curvlnear Regresson Monthly total producton costs and the number of unts produced at a local company over a perod of 10 months are shown below. Producton Costs (Y ) Unts Produced (X ) Month (n $ mllons) (n mllons) Z = X a. Usng Excel, enter the above data n a fle and save the fle. b. Draw a scatter dagram relatng X & Y. c. Perform a regresson and correlaton analyss relatng X & Y. d. Draw a scatter dagram relatng X & Y). e. If we can assume that a model n the form of: Y = β 0 + β 1 X + ε best descrbes the relatonshp between X and Y, Perform a regresson and correlaton analyss between X & Y. f. Compare the results of parts c and d and explan whch would be a better model and why? Professor Ahmad s Lecture Notes Page 11

12 Chapter 16 Problem Multple Regresson & Correlaton Wth Dummy Varables Fll n the Blanks Ahmad, Inc. s a mcrocomputer producer. The followng data represent Ahmad's yearly sales volume, ther advertsng expendture, and whether n a gven year they used all Televson advertsng (X = 0) or used Multmeda advertsng (X = 1). (Y) X1 X Sales Advertsng Dummy Varable Year ($1,000,000) ($10,000) (0,1) Regresson procedure of Excel was used on the above data and parts of the results are shown on the next page. a. Fll n all the blanks on the next page. b. Wrte the estmated regresson equaton. c. Usng the results shown on the next page, predct sales for the year 004 assumng we are plannng to use $700,000 for televson advertsng only. d. Usng the results shown on the next page, predct sales for the year 004 assumng we are plannng to use $700,000 for multmeda advertsng. Professor Ahmad s Lecture Notes Page 1

13 SUMMARY OUTPUT Multple R? R Square? Adjusted R Square? Standard Error.715 Observatons? ANOVA df SS MS F Sgnfcance F Regresson? ?? 8.59E-08 Resdual??? Total?? Coeffcents Standard Error t Stat P-value Intercept ?? Advertsng ?? Dummy ?? Professor Ahmad s Lecture Notes Page 13

14 Your Turn One Fnal Example Sgnfcance of Varables and Other Issues 3. Ahmad, Inc. produces several models of computer prnters. Data on a few varables for one of the company s prnters are presented below. Sales (Y) (In $1,000,000) Compettor's Prce (X3) (In $100) Advertsng (X1) (In $1,000) Prce (X) (In $100) Tme (X4) (In Years) Ratng (X5) (0 to 10) a. Enter the above data nto an Excel fle and save the fle. Prnt the fle and the results of all of the followng parts. b. Run a correlaton analyss (among all varables) and prnt the results. Fully dscuss the meanng of the correlaton coeffcents. Be sure to dscuss the concept of multcollnearty. c. Run a regresson analyss relatng sales (Y) and ALL the ndependent varables. Fully explan the results. d. Drop the varable(s) that at 95% confdence were not sgnfcant n part c and run a new regresson analyss. Fully explan your results. Professor Ahmad s Lecture Notes Page 14

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