CHAPTER 13. Simple Linear Regression LEARNING OBJECTIVES. USING Sunflowers Apparel

Transcription

1 CHAPTER 3 Smple Lear Regresso USING Suflowers Apparel 3 TYPES OF REGRESSION MODELS 3 DETERMINING THE SIMPLE LINEAR REGRESSION EQUATION The Least-Squares Method Vsual Exploratos: Explorg Smple Lear Regresso Coeffcets Predctos Regresso Aalyss: Iterpolato Versus Extrapolato Computg the Y Itercept, b 0, ad the Slope, b 33 MEASURES OF VARIATION Computg the Sum of Squares The Coeffcet of Determato Stadard Error of the Estmate 34 ASSUMPTIONS 35 RESIDUAL ANALYSIS Evaluatg the Assumptos 36 MEASURING AUTOCORRELATION: THE DURBIN-WATSON STATISTIC Resdual Plots to Detect Autocorrelato The Durb-Watso Statstc 37 INFERENCES ABOUT THE SLOPE AND CORRELATION COEFFICIENT t Test for the Slope F Test for the Slope Cofdece Iterval Estmate of the Slope (β ) t Test for the Correlato Coeffcet 38 ESTIMATION OF MEAN VALUES AND PREDICTION OF INDIVIDUAL VALUES The Cofdece Iterval Estmate The Predcto Iterval 39 PITFALLS IN REGRESSION AND ETHICAL ISSUES EXCEL COMPANION TO CHAPTER 3 E3 Performg Smple Lear Regresso Aalyses E3 Creatg Scatter Plots ad Addg a Predcto Le E33 Performg Resdual Aalyses E34 Computg the Durb-Watso Statstc E35 Estmatg the Mea of Y ad Predctg Y Values E36 Example: Suflowers Apparel Data LEARNING OBJECTIVES I ths chapter, you lear: To use regresso aalyss to predct the value of a depedet varable based o a depedet varable The meag of the regresso coeffcets b 0 ad b To evaluate the assumptos of regresso aalyss ad kow what to do f the assumptos are volated To make fereces about the slope ad correlato coeffcet To estmate mea values ad predct dvdual values

2 LEVIMC3_034057QXD 5 //07 4:4 PM Page 5 CHAPTER THIRTEEN Smple Lear Regresso Usg Suflowers Apparel The sales for Suflowers Apparel, a cha of upscale clothg stores for wome, have creased durg the past years as the cha has expaded the umber of stores ope Utl ow, Suflowers maagers selected stes based o subjectve factors, such as the avalablty of a good lease or the percepto that a locato seemed deal for a apparel store As the ew drector of plag, you eed to develop a systematc approach that wll lead to makg better decsos durg the ste selecto process As a startg pot, you beleve that the sze of the store sgfcatly cotrbutes to store sales, ad you wat to use ths relatoshp the decso-makg process How ca you use statstcs so that you ca forecast the aual sales of a proposed store based o the sze of that store? ths chapter ad the ext two chapters, you lear how regresso aalyss eables you to develop a model to predct the values of a umercal varable, based o the value of other varables I regresso aalyss, the varable you wsh to predct s called the depedet varable The varables used to make the predcto are called depedet varables I addto to predctg values of the depedet varable, regresso aalyss also allows you to detfy the type of mathematcal relatoshp that exsts betwee a depedet ad a depedet varable, to quatfy the effect that chages the depedet varable have o the depedet varable, ad to detfy uusual observatos For example, as the drector of plag, you may wsh to predct sales for a Suflowers store, based o the sze of the store Other examples clude predctg the mothly ret of a apartmet, based o ts sze, ad predctg the mothly sales of a product a supermarket, based o the amout of shelf space devoted to the product Ths chapter dscusses smple lear regresso, whch a sgle umercal depedet varable, X, s used to predct the umercal depedet varable Y, such as usg the sze of a store to predct the aual sales of the store Chapters 4 ad 5 dscuss multple regresso models, whch use several depedet varables to predct a umercal depedet varable, Y For example, you could use the amout of advertsg expedtures, prce, ad the amout of shelf space devoted to a product to predct ts mothly sales I 3 TYPES OF REGRESSION MODELS I Secto 5, you used a scatter plot (also kow as a scatter dagram) to exame the relatoshp betwee a X varable o the horzotal axs ad a Y varable o the vertcal axs The ature of the relatoshp betwee two varables ca take may forms, ragg from smple to extremely complcated mathematcal fuctos The smplest relatoshp cossts of a straghtle, or lear relatoshp A example of ths relatoshp s show Fgure 3

3 3: Types of Regresso Models 53 FIGURE 3 A postve straght-le relatoshp β 0 Y Y = chage Y X = chage X 0 X 0 Equato (3) represets the straght-le (lear) model SIMPLE LINEAR REGRESSION MODEL Y = β 0 + β X + ε (3) where β 0 = Y tercept for the populato β = slope for the populato ε = radom error Y for observato Y = depedet varable (sometmes referred to as the respose varable) for observato X = depedet varable (sometmes referred to as the explaatory varable) for observato The porto Y = β 0 + β X of the smple lear regresso model expressed Equato (3) s a straght le The slope of the le, β, represets the expected chage Y per ut chage X It represets the mea amout that Y chages (ether postvely or egatvely) for a oe-ut chage X The Y tercept, β 0, represets the mea value of Y whe X equals 0 The last compoet of the model, ε, represets the radom error Y for each observato, I other words, ε s the vertcal dstace of the actual value of Y above or below the predcted value of Y o the le The selecto of the proper mathematcal model depeds o the dstrbuto of the X ad Y values o the scatter plot I Pael A of Fgure 3 o page 54, the values of Y are geerally creasg learly as X creases Ths pael s smlar to Fgure 33 o page 55, whch llustrates the postve relatoshp betwee the square footage of the store ad the aual sales at braches of the Suflowers Apparel wome s clothg store cha Pael B s a example of a egatve lear relatoshp As X creases, the values of Y are geerally decreasg A example of ths type of relatoshp mght be the prce of a partcular product ad the amout of sales The data Pael C show a postve curvlear relatoshp betwee X ad Y The values of Y crease as X creases, but ths crease tapers off beyod certa values of X A example of a postve curvlear relatoshp mght be the age ad mateace cost of a mache As a mache gets older, the mateace cost may rse rapdly at frst, but the level off beyod a certa umber of years Pael D shows a U-shaped relatoshp betwee X ad Y As X creases, at frst Y geerally decreases; but as X cotues to crease, Y ot oly stops decreasg but actually creases above ts mmum value A example of ths type of relatoshp mght be the umber of errors per hour at a task ad the umber of hours worked The umber of errors per hour

4 54 CHAPTER THIRTEEN Smple Lear Regresso FIGURE 3 Examples of types of relatoshps foud scatter plots Y Y Pael A Postve lear relatoshp X Pael B Negatve lear relatoshp X Y Y X Pael C Postve curvlear relatoshp Y X Pael D U-shaped curvlear relatoshp Y X Pael E Negatve curvlear relatoshp X Pael F No relatoshp betwee X ad Y decreases as the dvdual becomes more profcet at the task, but the t creases beyod a certa pot because of factors such as fatgue ad boredom Pael E dcates a expoetal relatoshp betwee X ad Y I ths case, Y decreases very rapdly as X frst creases, but the t decreases much less rapdly as X creases further A example of a expoetal relatoshp could be the resale value of a automoble ad ts age I the frst year, the resale value drops drastcally from ts orgal prce; however, the resale value the decreases much less rapdly subsequet years Fally, Pael F shows a set of data whch there s very lttle or o relatoshp betwee X ad Y Hgh ad low values of Y appear at each value of X I ths secto, a varety of dfferet models that represet the relatoshp betwee two varables were brefly examed Although scatter plots are useful vsually dsplayg the mathematcal form of a relatoshp, more sophstcated statstcal procedures are avalable to determe the most approprate model for a set of varables The rest of ths chapter dscusses the model used whe there s a lear relatoshp betwee varables 3 DETERMINING THE SIMPLE LINEAR REGRESSION EQUATION I the Usg Statstcs scearo o page 5, the stated goal s to forecast aual sales for all ew stores, based o store sze To exame the relatoshp betwee the store sze square feet ad ts aual sales, a sample of 4 stores was selected Table 3 summarzes the results for these 4 stores, whch are stored the fle stexls

5 3: Determg the Smple Lear Regresso Equato 55 TABLE 3 Square Footage ( Thousads of Square Feet) ad Aual Sales ( Mllos of Dollars) for a Sample of 4 Braches of Suflowers Apparel Square Aual Sales Square Aual Sales Feet ( Mllos Feet ( Mllos Store (Thousads) of Dollars) Store (Thousads) of Dollars) Fgure 33 dsplays the scatter plot for the data Table 3 Observe the creasg relatoshp betwee square feet (X) ad aual sales (Y) As the sze of the store creases, aual sales crease approxmately as a straght le Thus, you ca assume that a straght le provdes a useful mathematcal model of ths relatoshp Now you eed to determe the specfc straght le that s the best ft to these data FIGURE 33 Mcrosoft Excel scatter plot for the Suflowers Apparel data See Secto E to create ths The Least-Squares Method I the precedg secto, a statstcal model s hypotheszed to represet the relatoshp betwee two varables, square footage ad sales, the etre populato of Suflowers Apparel stores However, as show Table 3, the data are from oly a radom sample of stores If certa assumptos are vald (see Secto 34), you ca use the sample Y tercept, b 0, ad the sample slope, b, as estmates of the respectve populato parameters, β 0 ad β Equato (3) uses these estmates to form the smple lear regresso equato Ths straght le s ofte referred to as the predcto le SIMPLE LINEAR REGRESSION EQUATION: THE PREDICTION LINE The predcted value of Y equals the Y tercept plus the slope tmes the value of X Yˆ = b + b X 0 (3)

6 56 CHAPTER THIRTEEN Smple Lear Regresso where Yˆ = predcted value of Y for observato X = value of X for observato b 0 = sample Y tercept b = sample slope Equato (3) requres the determato of two regresso coeffcets b 0 (the sample Y tercept) ad b (the sample slope) The most commo approach to fdg b 0 ad b s the method of least squares Ths method mmzes the sum of the squared dffereces betwee the actual values (Y ) ad the predcted values ( Yˆ ) usg the smple lear regresso equato [that s, the predcto le; see Equato (3)] Ths sum of squared dffereces s equal to ( Y Yˆ ) Because Yˆ = b + b X, 0 0 ( Y Yˆ ) = [ Y ( b + b X )] Because ths equato has two ukows, b 0 ad b, the sum of squared dffereces depeds o the sample Y tercept, b 0, ad the sample slope, b The least-squares method determes the values of b 0 ad b that mmze the sum of squared dffereces Ay values for b 0 ad b other tha those determed by the least-squares method result a greater sum of squared dffereces betwee the actual values (Y ) ad the predcted values Yˆ I ths book, Mcrosoft Excel s used to perform the computatos volved the least-squares method For the data of Table 3, Fgure 34 presets results from Mcrosoft Excel FIGURE 34 Mcrosoft Excel results for the Suflowers Apparel data See Secto E3 to create ths S YX SSR SSE SST p-value b 0 b

7 3: Determg the Smple Lear Regresso Equato 57 To uderstad how the results are computed, may of the computatos volved are llustrated Examples 33 ad 34 o pages 50 5 ad I Fgure 34, observe that b 0 = ad b = 6699 Thus, the predcto le [see Equato (3) o page 55] for these data s Yˆ = X The slope, b, s Ths meas that for each crease of ut X, the mea value of Y s estmated to crease by 6699 uts I other words, for each crease of 0 thousad square feet the sze of the store, the mea aual sales are estmated to crease by 6699 mllos of dollars Thus, the slope represets the porto of the aual sales that are estmated to vary accordg to the sze of the store The Y tercept, b 0, s The Y tercept represets the mea value of Y whe X equals 0 Because the square footage of the store caot be 0, ths Y tercept has o practcal terpretato Also, the Y tercept for ths example s outsde the rage of the observed values of the X varable, ad therefore terpretatos of the value of b 0 should be made cautously Fgure 35 dsplays the actual observatos ad the predcto le To llustrate a stuato whch there s a drect terpretato for the Y tercept, b 0, see Example 3 FIGURE 35 Mcrosoft Excel scatter plot ad predcto le for Suflowers Apparel data See Secto E3 to create ths EXAMPLE 3 INTERPRETING THE Y INTERCEPT, b 0, AND THE SLOPE, b A statstcs professor wats to use the umber of hours a studet studes for a statstcs fal exam (X) to predct the fal exam score (Y) A regresso model was ft based o data collected for a class durg the prevous semester, wth the followg results: Yˆ = X What s the terpretato of the Y tercept, b 0, ad the slope, b? SOLUTION The Y tercept b 0 = 350 dcates that whe the studet does ot study for the fal exam, the mea fal exam score s 350 The slope b = 3 dcates that for each crease of oe hour studyg tme, the mea chage the fal exam score s predcted to be +30 I other words, the fal exam score s predcted to crease by 3 pots for each oe-hour crease studyg tme

8 58 CHAPTER THIRTEEN Smple Lear Regresso VISUAL EXPLORATIONS Explorg Smple Lear Regresso Coeffcets Use the Vsual Exploratos Smple Lear Regresso procedure to produce a predcto le that s as close as possble to the predcto le defed by the least-squares soluto Ope the Vsual Exploratosxla add- workbook ad select VsualExploratos Smple Lear Regresso (Excel ) or Add-s Vsual Exploratos Smple Lear Regresso (Excel 007) (See Secto E6 to lear about usg add-s) Whe a scatter plot of the Suflowers Apparel data of Table 3 o page 55 wth a tal predcto le appears (show below), clck the sper buttos to chage the values for b, the slope of the predcto le, ad b 0, the Y tercept of the predcto le Try to produce a predcto le that s as close as possble to the predcto le defed by the least-squares estmates, usg the chart dsplay ad the Dfferece from Target SSE value as feedback (see page 55 for a explaato of SSE) Clck Fsh whe you are doe wth ths explorato At ay tme, clck Reset to reset the b ad b 0 values, Help for more formato, or Soluto to reveal the predcto le defed by the least-squares method workbook ad select VsualExploratos Smple Lear Regresso wth your worksheet data (97-003) or Add-s Vsual Exploratos Smple Lear Regresso wth your worksheet data (007) I the procedure s dalog box (show below), eter your Y varable cell rage as the Y Varable Cell Rage ad your X varable cell rage as the X Varable Cell Rage Clck Frst cells both rages cota a label, eter a ttle as the Ttle, ad clck OK Whe the scatter plot wth a tal predcto le appears, use the structos the frst part of ths secto to try to produce the predcto le defed by the least-squares method Usg Your Ow Regresso Data To use Vsual Exploratos to fd a predcto le for your ow data, ope the Vsual Exploratosxla add-

9 3: Determg the Smple Lear Regresso Equato 59 Retur to the Usg Statstcs scearo cocerg the Suflowers Apparel stores Example 3 llustrates how you use the predcto equato to predct the mea aual sales EXAMPLE 3 PREDICTING MEAN ANNUAL SALES, BASED ON SQUARE FOOTAGE Use the predcto le to predct the mea aual sales for a store wth 4,000 square feet SOLUTION You ca determe the predcted value by substtutg X = 4 (thousads of square feet) to the smple lear regresso equato: Yˆ = X Yˆ = ( 4) = or $ 7, 644, 000 Thus, the predcted mea aual sales of a store wth 4,000 square feet s $7,644,000 Predctos Regresso Aalyss: Iterpolato Versus Extrapolato Whe usg a regresso model for predcto purposes, you eed to cosder oly the relevat rage of the depedet varable makg predctos Ths relevat rage cludes all values from the smallest to the largest X used developg the regresso model Hece, whe predctg Y for a gve value of X, you ca terpolate wth ths relevat rage of the X values, but you should ot extrapolate beyod the rage of X values Whe you use the square footage to predct aual sales, the square footage ( thousads of square feet) vares from to 58 (see Table 3 o page 55) Therefore, you should predct aual sales oly for stores whose sze s betwee ad 58 thousads of square feet Ay predcto of aual sales for stores outsde ths rage assumes that the observed relatoshp betwee sales ad store sze for store szes from to 58 thousad square feet s the same as for stores outsde ths rage For example, you caot extrapolate the lear relatoshp beyod 5,800 square feet Example 3 It would be mproper to use the predcto le to forecast the sales for a ew store cotag 8,000 square feet It s qute possble that store sze has a pot of dmshg returs If that s true, as square footage creases beyod 5,800 square feet, the effect o sales mght become smaller ad smaller Computg the Y Itercept, b 0, ad the Slope, b For small data sets, you ca use a had calculator to compute the least-squares regresso coeffcets Equatos (33) ad (34) gve the values of b 0 ad b, whch mmze 0 ( Y Yˆ ) = [ Y ( b + b X )] COMPUTATIONAL FORMULA FOR THE SLOPE, b where SSXY = SSX = ( X X) = X ( X X)( Y Y) = X Y b = SSXY SSX X X (33) Y

10 50 CHAPTER THIRTEEN Smple Lear Regresso COMPUTATIONAL FORMULA FOR THE Y INTERCEPT, b 0 b0 = Y bx (34) where Y = Y X = X EXAMPLE 33 COMPUTING THE Y INTERCEPT, b 0, AND THE SLOPE, b Compute the Y tercept, b 0, ad the slope, b, for the Suflowers Apparel data SOLUTION Examg Equatos (33) ad (34), you see that fve quattes must be calculated to determe b ad b 0 These are, the sample sze; Y X, the sum of the X values;, the sum of the Y values;, the sum of the squared X values; ad XY, the sum of the product of X ad Y For the Suflowers Apparel data, the umber of square feet s used to predct the aual sales a store Table 3 presets the computatos of the varous sums eeded for the ste selecto problem, plus used to compute SST Secto 33 Y X, the sum of the squared Y values that wll be TABLE 3 Computatos for the Suflowers Apparel Data Square Aual Store Feet (X ) Sales (Y ) X Y XY Totals

11 3: Determg the Smple Lear Regresso Equato 5 Usg Equatos (33) ad (34), you ca compute the values of b 0 ad b : b = SSXY SSX SSXY = ( X X )( Y Y ) = X Y X Y ( 40 9)( 8 8) SSXY = = = SSX = ( X X ) = X ( 40 9) = = = X so that b = = 6699 ad b = Y b X Y X b 0 0 Y 8 8 = = = X 40 9 = = = = ( 6699)( 943) =

12 5 CHAPTER THIRTEEN Smple Lear Regresso PROBLEMS FOR SECTION 3 Learg the Bascs PH Grade ASSIST 3 Fttg a straght le to a set of data yelds the followg predcto le: a Iterpret the meag of the Y tercept, b 0 b Iterpret the meag of the slope, b c Predct the mea value of Y for X = 3 3 If the values of X Problem 3 rage from to 5, should you use ths model to predct the mea value of Y whe X equals a 3? b 3? c 0? d 4? PH Grade ASSIST 33 Fttg a straght le to a set of data yelds the followg predcto le: Yˆ = 6 0 5X a Iterpret the meag of the Y tercept, b 0 b Iterpret the meag of the slope, b c Predct the mea value of Y for X = 6 Applyg the Cocepts PH Grade ASSIST SELF Test Yˆ 34 The marketg maager of a large supermarket cha would lke to use shelf space to predct the sales of pet food A radom sample of equal-szed stores s selected, wth the followg results (stored the fle petfoodxls): = + 5X Shelf Space (X ) Weekly Sales (Y ) Store (Feet) ($) a Costruct a scatter plot For these data, b 0 = 45 ad b = 74 b Iterpret the meag of the slope, b, ths problem c Predct the mea weekly sales ( hudreds of dollars) of pet food for stores wth 8 feet of shelf space for pet food 35 Crculato s the lfeblood of the publshg busess The larger the sales of a magaze, the more t ca charge advertsers Recetly, a crculato gap has appeared betwee the publshers reports of magazes ewsstad sales ad subsequet audts by the Audt Bureau of Crculatos The data the fle crculatoxls represet the reported ad audted ewsstad sales ( thousads) 00 for the followg 0 magazes: Magaze Reported (X ) Audted (Y ) YM CosmoGrl Rose Playboy Esqure TeePeople More 55 9 Sp Vogue Elle Source: Extracted from M Rose, I Fght for Ads, Publshers Ofte Overstate Ther Sales, The Wall Street Joural, August 6, 003, pp A, A0 a Costruct a scatter plot For these data b 0 = 674 ad b = 0579 b Iterpret the meag of the slope, b, ths problem c Predct the mea audted ewsstad sales for a magaze that reports ewsstad sales of 400, The ower of a movg compay typcally has hs most expereced maager predct the total umber of labor hours that wll be requred to complete a upcomg move Ths approach has proved useful the past, but he would lke to be able to develop a more accurate method of predctg labor hours by usg the umber of cubc feet moved I a prelmary effort to provde a more accurate method, he has collected data for 36 moves whch the org ad destato were wth the borough of Mahatta New York Cty ad whch the travel tme was a sgfcat porto of the hours worked The data are stored the fle movgxls

13 3: Determg the Smple Lear Regresso Equato 53 a Costruct a scatter plot b Assumg a lear relatoshp, use the least-squares method to fd the regresso coeffcets b 0 ad b c Iterpret the meag of the slope, b, ths problem d Predct the mea labor hours for movg 500 cubc feet PH Grade ASSIST 37 A large mal-order house beleves that there s a lear relatoshp betwee the weght of the mal t receves ad the umber of orders to be flled It would lke to vestgate the relatoshp order to predct the umber of orders, based o the weght of the mal From a operatoal perspectve, kowledge of the umber of orders wll help the plag of the orderfulfllmet process A sample of 5 mal shpmets s selected that rage from 00 to 700 pouds The results (stored the fle malxls) are as follows: Weght Weght of Mal Orders of Mal Orders (Pouds) (Thousads) (Pouds) (Thousads) a Costruct a scatter plot b Assumg a lear relatoshp, use the least-squares method to fd the regresso coeffcets b 0 ad b c Iterpret the meag of the slope, b, ths problem d Predct the mea umber of orders whe the weght of the mal s 500 pouds 38 The value of a sports frachse s drectly related to the amout of reveue that a frachse ca geerate The data the fle bbreveuexls represet the value 005 ( mllos of dollars) ad the aual reveue ( mllos of dollars) for 30 baseball frachses Suppose you wat to develop a smple lear regresso model to predct frachse value based o aual reveue geerated a Costruct a scatter plot b Use the least-squares method to fd the regresso coeffcets b 0 ad b c Iterpret the meag of b 0 ad b ths problem d Predct the mea value of a baseball frachse that geerates $50 mllo of aual reveue 39 A aget for a resdetal real estate compay a large cty would lke to be able to predct the mothly retal cost for apartmets, based o the sze of the apartmet, as defed by square footage A sample of 5 apartmets (stored the fle retxls) a partcular resdetal eghborhood was selected, ad the formato gathered revealed the followg: Mothly Sze Mothly Sze Ret (Square Ret (Square Apartmet ($) Feet) Apartmet ($) Feet) ,800,369,600,450 5,400,75 3,00,085 6,450,5 4,500,3 7,00, ,700,59 6,700,485 9,00,50 7,650,36 0, ,600, ,650,040 0, ,00 755,400, ,000,650,85 5,750,00 3,300,985 a Costruct a scatter plot b Use the least-squares method to fd the regresso coeffcets b 0 ad b c Iterpret the meag of b 0 ad b ths problem d Predct the mea mothly ret for a apartmet that has,000 square feet e Why would t ot be approprate to use the model to predct the mothly ret for apartmets that have 500 square feet? f Your freds Jm ad Jefer are cosderg sgg a lease for a apartmet ths resdetal eghborhood They are tryg to decde betwee two apartmets, oe wth,000 square feet for a mothly ret of $,75 ad the other wth,00 square feet for a mothly ret of $,45 What would you recommed to them based o (a) through (d)? 30 The data the fle hardessxls provde measuremets o the hardess ad tesle stregth for 35 specmes of de-cast alumum It s beleved that hardess (measured Rockwell E uts) ca be used to predct tesle stregth (measured thousads of pouds per square ch) a Costruct a scatter plot b Assumg a lear relatoshp, use the least-squares method to fd the regresso coeffcets b 0 ad b c Iterpret the meag of the slope, b, ths problem d Predct the mea tesle stregth for de-cast alumum that has a hardess of 30 Rockwell E uts

14 54 CHAPTER THIRTEEN Smple Lear Regresso 33 MEASURES OF VARIATION Whe usg the least-squares method to determe the regresso coeffcets for a set of data, you eed to compute three mportat measures of varato The frst measure, the total sum of squares (SST ), s a measure of varato of the Y values aroud ther mea, Y I a regresso aalyss, the total varato or total sum of squares s subdvded to explaed varato ad uexplaed varato The explaed varato or regresso sum of squares (SSR) s due to the relatoshp betwee X ad Y, ad the uexplaed varato, or error sum of squares (SSE) s due to factors other tha the relatoshp betwee X ad Y Fgure 36 shows these dfferet measures of varato FIGURE 36 Measures of varato Y Y Error sum of squares (Y Y ) = SSE = Y = b 0 + b X Total sum of squares (Y Y) = SST = Regresso sum of squares (Y Y) = SSR = Y 0 X X Computg the Sum of Squares The regresso sum of squares (SSR) s based o the dfferece betwee Yˆ (the predcted value of Y from the predcto le ) ad Y (the mea value of Y) The error sum of squares (SSE) represets the part of the varato Y that s ot explaed by the regresso It s based o the dfferece betwee Y ad Yˆ Equatos (35), (36), (37), ad (38) defe these measures of varato MEASURES OF VARIATION IN REGRESSION The total sum of squares s equal to the regresso sum of squares plus the error sum of squares SST = SSR + SSE (35) TOTAL SUM OF SQUARES (SST) The total sum of squares (SST) s equal to the sum of the squared dffereces betwee each observed Y value ad Y, the mea value of Y SST = Total sum of squares = ( Y Y ) (36)

15 33: Measures of Varato 55 REGRESSION SUM OF SQUARES (SSR) The regresso sum of squares (SSR) s equal to the sum of the squared dffereces betwee the predcted value of Y ad Y, the mea value of Y SSR = Explaed varato or regresso of squares = ( Yˆ Y ) (37) ERROR SUM OF SQUARES (SSE) The error sum of squares (SSE) s equal to the sum of the squared dffereces betwee the observed value of Y ad the predcted value of Y SSE = Uexplaed varato or error sum of squares = ( Y Yˆ ) (38) Fgure 37 shows the sum of squares area of the worksheet cotag the Mcrosoft Excel results for the Suflowers Apparel data The total varato, SST, s equal to Ths amout s subdvded to the sum of squares explaed by the regresso (SSR), equal to , ad the sum of squares uexplaed by the regresso (SSE), equal to 067 From Equato (35) o page 54: SST = SSR + SSE = FIGURE 37 Mcrosoft Excel sum of squares for the Suflowers Apparel data See Secto E3 to create the worksheet that cotas ths area I a data set that has a large umber of sgfcat dgts, the results of a regresso aalyss are sometmes dsplayed usg a umercal format kow as scetfc otato Ths type of format s used to dsplay very small or very large values The umber after the letter E represets the umber of dgts that the decmal pot eeds to be moved to the left (for a egatve umber) or to the rght (for a postve umber) For example, the umber 3743E+0 meas that the decmal pot should be moved two places to the rght, producg the umber 3743 The umber 3743E-0 meas that the decmal pot should be moved two places to the left, producg the umber Whe scetfc otato s used, fewer sgfcat dgts are usually dsplayed, ad the umbers may appear to be rouded

16 56 CHAPTER THIRTEEN Smple Lear Regresso The Coeffcet of Determato By themselves, SSR, SSE, ad SST provde lttle formato However, the rato of the regresso sum of squares (SSR) to the total sum of squares (SST ) measures the proporto of varato Y that s explaed by the depedet varable X the regresso model Ths rato s called the coeffcet of determato, r, ad s defed Equato (39) COEFFICIENT OF DETERMINATION The coeffcet of determato s equal to the regresso sum of squares (that s, explaed varato) dvded by the total sum of squares (that s, total varato) r Regresso sum of squares = = Total sum of squares SSR SST (39) The coeffcet of determato measures the proporto of varato Y that s explaed by the depedet varable X the regresso model For the Suflowers Apparel data, wth SSR = , SSE = 067, ad SST = 69543, r = = Therefore, 904% of the varato aual sales s explaed by the varablty the sze of the store, as measured by the square footage Ths large r dcates a strog postve lear relatoshp betwee two varables because the use of a regresso model has reduced the varablty predctg aual sales by 904% Oly 958% of the sample varablty aual sales s due to factors other tha what s accouted for by the lear regresso model that uses square footage Fgure 38 presets the coeffcet of determato porto of the Mcrosoft Excel results for the Suflowers Apparel data FIGURE 38 Partal Mcrosoft Excel regresso results for the Suflowers Apparel data S YX See Secto E3 to create the worksheet that cotas ths area EXAMPLE 34 COMPUTING THE COEFFICIENT OF DETERMINATION Compute the coeffcet of determato, r, for the Suflowers Apparel data SOLUTION You ca compute SST, SSR, ad SSE, that are defed Equatos (36), (37), ad (38) o pages 54 55, by usg Equatos (30), (3), ad (3) COMPUTATIONAL FORMULA FOR SST Y SST = ( Y Y ) = Y (30)

17 33: Measures of Varato 57 COMPUTATIONAL FORMULA FOR SSR SSR = ( Yˆ Y ) = b0 Y + b XY Y (3) COMPUTATIONAL FORMULA FOR SSE SSE = ( Y Yˆ) = Y b0 Y b X Y (3) Usg the summary results from Table 3 o page 50, SST = ( Yˆ Y ) = Y SSR = ( Yˆ Y ) ( 8 8) = = = = b Y + b X Y 0 0 Y Y ( 8 8) = ( )( 8 8) + ( 66986)( 30 3) 4 = SSE = ( Y Yˆ ) = Y b Y b X Y = ( )( 8 8) ( 66986)( 30 3) = 067 Therefore, r = =

18 58 CHAPTER THIRTEEN Smple Lear Regresso Stadard Error of the Estmate Although the least-squares method results the le that fts the data wth the mmum amout of error, uless all the observed data pots fall o a straght le, the predcto le s ot a perfect predctor Just as all data values caot be expected to be exactly equal to ther mea, ether ca they be expected to fall exactly o the predcto le A mportat statstc, called the stadard error of the estmate, measures the varablty of the actual Y values from the predcted Y values the same way that the stadard devato Chapter 3 measures the varablty of each value aroud the sample mea I other words, the stadard error of the estmate s the stadard devato aroud the predcto le, whereas the stadard devato Chapter 3 s the stadard devato aroud the sample mea Fgure 35 o page 57 llustrates the varablty aroud the predcto le for the Suflowers Apparel data Observe that although may of the actual values of Y fall ear the predcto le, oe of the values are exactly o the le The stadard error of the estmate, represeted by the symbol S YX, s defed Equato (33) STANDARD ERROR OF THE ESTIMATE S YX = ( Y Yˆ ) SSE = (33) where Y = actual value of Y for a gve X ˆ Y = predcted value of Y for a gve X SSE = error sum of squares From Equato (38) ad Fgure 34 o page 56, SSE = 067 Thus, S YX = = Ths stadard error of the estmate, equal to mllos of dollars (that s, $966,400), s labeled Stadard Error the Mcrosoft Excel results show Fgure 38 o page 56 The stadard error of the estmate represets a measure of the varato aroud the predcto le It s measured the same uts as the depedet varable Y The terpretato of the stadard error of the estmate s smlar to that of the stadard devato Just as the stadard devato measures varablty aroud the mea, the stadard error of the estmate measures varablty aroud the predcto le For Suflowers Apparel, the typcal dfferece betwee actual aual sales at a store ad the predcted aual sales usg the regresso equato s approxmately $966,400 PROBLEMS FOR SECTION 33 Learg the Bascs PH Grade ASSIST PH Grade ASSIST 3 How do you terpret a coeffcet of determato, r, equal to 080? 3 If SSR = 36 ad SSE = 4, determe SST ad the compute the coeffcet of determato, r, ad terpret ts meag PH Grade ASSIST PH Grade ASSIST 33 If SSR = 66 ad SST = 88, compute the coeffcet of determato, r, ad terpret ts meag 34 If SSE = 0 ad SSR = 30, compute the coeffcet of determato, r, ad terpret ts meag

19 34: Assumptos If SSR = 0, why s t mpossble for SST to equal 0? Applyg the Cocepts PH Grade ASSIST SELF Test 36 I Problem 34 o page 5, the marketg maager used shelf space for pet food to predct weekly sales (stored the fle petfoodxls) For that data, SSR = 0,535 ad SST = 30,05 a Determe the coeffcet of determato, r, ad terpret ts meag b Determe the stadard error of the estmate c How useful do you thk ths regresso model s for predctg sales? 37 I Problem 35 o page 5, you used reported magaze ewsstad sales to predct audted sales (stored the fle crculatoxls) For that data, SSR = 30,304 ad SST = 44,53864 a Determe the coeffcet of determato, r, ad terpret ts meag b Determe the stadard error of the estmate c How useful do you thk ths regresso model s for predctg audted sales? 38 I Problem 36 o page 5, a ower of a movg compay wated to predct labor hours, based o the cubc feet moved (stored the fle movgxls) Usg the results of that problem, a determe the coeffcet of determato, r, ad terpret ts meag b determe the stadard error of the estmate c How useful do you thk ths regresso model s for predctg labor hours? PH Grade ASSIST 39 I Problem 37 o page 53, you used the weght of mal to predct the umber of orders receved (stored the fle malxls) Usg the results of that problem, a determe the coeffcet of determato, r, ad terpret ts meag b fd the stadard error of the estmate c How useful do you thk ths regresso model s for predctg the umber of orders? 30 I Problem 38 o page 53, you used aual reveues to predct the value of a baseball frachse (stored the fle bbreveuexls) Usg the results of that problem, a determe the coeffcet of determato, r, ad terpret ts meag b determe the stadard error of the estmate c How useful do you thk ths regresso model s for predctg the value of a baseball frachse? 3 I Problem 39 o page 53, a aget for a real estate compay wated to predct the mothly ret for apartmets, based o the sze of the apartmet (stored the fle retxls) Usg the results of that problem, a determe the coeffcet of determato, r, ad terpret ts meag b determe the stadard error of the estmate c How useful do you thk ths regresso model s for predctg the mothly ret? 3 I Problem 30 o page 53, you used hardess to predct the tesle stregth of de-cast alumum (stored the fle hardessxls) Usg the results of that problem, a determe the coeffcet of determato, r, ad terpret ts meag b fd the stadard error of the estmate c How useful do you thk ths regresso model s for predctg the tesle stregth of de-cast alumum? 34 ASSUMPTIONS The dscusso of hypothess testg ad the aalyss of varace emphaszed the mportace of the assumptos to the valdty of ay coclusos reached The assumptos ecessary for regresso are smlar to those of the aalyss of varace because both topcs fall the geeral category of lear models (referece 4) The four assumptos of regresso (kow by the acroym LINE) are as follows: Learty Idepedece of errors Normalty of error Equal varace The frst assumpto, learty, states that the relatoshp betwee varables s lear Relatoshps betwee varables that are ot lear are dscussed Chapter 5 The secod assumpto, depedece of errors, requres that the errors (ε ) are depedet of oe aother Ths assumpto s partcularly mportat whe data are collected over a perod of tme I such stuatos, the errors for a specfc tme perod are sometmes correlated wth those of the prevous tme perod

20 530 CHAPTER THIRTEEN Smple Lear Regresso The thrd assumpto, ormalty, requres that the errors (ε ) are ormally dstrbuted at each value of X Lke the t test ad the ANOVA F test, regresso aalyss s farly robust agast departures from the ormalty assumpto As log as the dstrbuto of the errors at each level of X s ot extremely dfferet from a ormal dstrbuto, fereces about β 0 ad β are ot serously affected The fourth assumpto, equal varace or homoscedastcty, requres that the varace of the errors (ε ) are costat for all values of X I other words, the varablty of Y values s the same whe X s a low value as whe X s a hgh value The equal varace assumpto s mportat whe makg fereces about β 0 ad β If there are serous departures from ths assumpto, you ca use ether data trasformatos or weghted least-squares methods (see referece 4) 35 RESIDUAL ANALYSIS I Secto 3, regresso aalyss was troduced I Sectos 3 ad 33, a regresso model was developed usg the least-squares approach for the Suflowers Apparel data Is ths the correct model for these data? Are the assumptos troduced Secto 34 vald? I ths secto, a graphcal approach called resdual aalyss s used to evaluate the assumptos ad determe whether the regresso model selected s a approprate model The resdual or estmated error value, e, s the dfferece betwee the observed (Y ) ad predcted ( Yˆ ) values of the depedet varable for a gve value of X Graphcally, a resdual appears o a scatter plot as the vertcal dstace betwee a observed value of Y ad the predcto le Equato (34) defes the resdual RESIDUAL The resdual s equal to the dfferece betwee the observed value of Y ad the predcted value of Y e = Y Yˆ (34) Evaluatg the Assumptos Recall from Secto 34 that the four assumptos of regresso (kow by the acroym LINE) are learty, depedece, ormalty, ad equal varace Learty To evaluate learty, you plot the resduals o the vertcal axs agast the correspodg X values of the depedet varable o the horzotal axs If the lear model s approprate for the data, there s o apparet patter ths plot However, f the lear model s ot approprate, there s a relatoshp betwee the X values ad the resduals, e You ca see such a patter Fgure 39 Pael A shows a stuato whch, although there s a creasg tred Y as X creases, the relatoshp seems curvlear because the upward tred decreases for creasg values of X Ths quadratc effect s hghlghted Pael B, where there s a clear relatoshp betwee X ad e By plottg the resduals, the lear tred of X wth Y has bee removed, thereby exposg the lack of ft the smple lear model Thus, a quadratc model s a better ft ad should be used place of the smple lear model (See Secto 5 for further dscusso of fttg quadratc models) To determe whether the smple lear regresso model s approprate, retur to the evaluato of the Suflowers Apparel data Fgure 30 provdes the predcted ad resdual values of the respose varable (aual sales) computed by Mcrosoft Excel

21 35: Resdual Aalyss 53 FIGURE 39 Studyg the approprateess of the smple lear regresso model Y e 0 Pael A X Pael B X FIGURE 30 Mcrosoft Excel resdual statstcs for the Suflowers Apparel data See Secto E33 to create the worksheet that cotas ths area To assess learty, the resduals are plotted agast the depedet varable (store sze, thousads of square feet) Fgure 3 Although there s wdespread scatter the resdual plot, there s o apparet patter or relatoshp betwee the resduals ad X The resduals appear to be evely spread above ad below 0 for the dfferg values of X You ca coclude that the lear model s approprate for the Suflowers Apparel data FIGURE 3 Mcosoft Excel plot of resduals agast the square footage of a store for the Suflowers Apparel data See Secto E to create ths

22 53 CHAPTER THIRTEEN Smple Lear Regresso Idepedece You ca evaluate the assumpto of depedece of the errors by plottg the resduals the order or sequece whch the data were collected Data collected over perods of tme sometmes exhbt a autocorrelato effect amog successve observatos I these staces, there s a relatoshp betwee cosecutve resduals If ths relatoshp exsts (whch volates the assumpto of depedece), t s apparet the plot of the resduals versus the tme whch the data were collected You ca also test for autocorrelato by usg the Durb-Watso statstc, whch s the subject of Secto 36 Because the Suflowers Apparel data were collected durg the same tme perod, you do ot eed to evaluate the depedece assumpto Normalty You ca evaluate the assumpto of ormalty the errors by tallyg the resduals to a frequecy dstrbuto ad dsplayg the results a hstogram (see Secto 3) For the Suflowers Apparel data, the resduals have bee talled to a frequecy dstrbuto Table 33 (There are a suffcet umber of values, however, to costruct a hstogram) You ca also evaluate the ormalty assumpto by comparg the actual versus theoretcal values of the resduals or by costructg a ormal probablty plot of the resduals (see Secto 63) Fgure 3 s a ormal probablty plot of the resduals for the Suflower Apparel data TABLE 33 Frequecy Dstrbuto of 4 Resdual Values for the Suflowers Apparel Data Resduals Frequecy 5 but less tha but less tha but less tha but less tha but less tha but less tha but less tha FIGURE 3 Mcrosoft Excel ormal probablty plot of the resduals for the Suflowers Apparel data See Secto E6 to create ths It s dffcult to evaluate the ormalty assumpto for a sample of oly 4 values, regardless of whether you use a hstogram, stem-ad-leaf dsplay, box-ad-whsker plot, or ormal probablty plot You ca see from Fgure 3 that the data do ot appear to depart substatally from a ormal dstrbuto The robustess of regresso aalyss wth modest departures from ormalty eables you to coclude that you should ot be overly cocered about departures from ths ormalty assumpto the Suflowers Apparel data

23 35: Resdual Aalyss 533 Equal Varace You ca evaluate the assumpto of equal varace from a plot of the resduals wth X For the Suflowers Apparel data of Fgure 3 o page 53, there do ot appear to be major dffereces the varablty of the resduals for dfferet X values Thus, you ca coclude that there s o apparet volato the assumpto of equal varace at each level of X To exame a case whch the equal varace assumpto s volated, observe Fgure 33, whch s a plot of the resduals wth X for a hypothetcal set of data I ths plot, the varablty of the resduals creases dramatcally as X creases, demostratg the lack of homogeety the varaces of Y at each level of X For these data, the equal varace assumpto s vald FIGURE 33 Volato of equal varace Resduals 0 X PROBLEMS FOR SECTION 35 Learg the Bascs 33 The results below provde the X values, resduals, ad a resdual plot from a regresso aalyss: 34 The results below show the X values, resduals, ad a resdual plot from a regresso aalyss: Is there ay evdece of a patter the resduals? Expla Is there ay evdece of a patter the resduals? Expla

24 534 CHAPTER THIRTEEN Smple Lear Regresso Applyg the Cocepts 35 I Problem 35 o page 5, you used reported magaze ewsstad sales to predct audted sales The data are stored the fle crculatoxls Perform a resdual aalyss for these data a Determe the adequacy of the ft of the model b Evaluate whether the assumptos of regresso have bee serously volated SELF Test 36 I Problem 34 o page 5, the marketg maager used shelf space for pet food to predct weekly sales The data are stored the fle petfoodxls Perform a resdual aalyss for these data a Determe the adequacy of the ft of the model b Evaluate whether the assumptos of regresso have bee serously volated 37 I Problem 37 o page 53, you used the weght of mal to predct the umber of orders receved Perform a resdual aalyss for these data The data are stored the fle malxls Based o these results, a determe the adequacy of the ft of the model b evaluate whether the assumptos of regresso have bee serously volated 38 I Problem 36 o page 5, the ower of a movg compay wated to predct labor hours based o the cubc feet moved Perform a resdual aalyss for these data The data are stored the fle movgxls Based o these results, a determe the adequacy of the ft of the model b evaluate whether the assumptos of regresso have bee serously volated 39 I Problem 39 o page 53, a aget for a real estate compay wated to predct the mothly ret for apartmets, based o the sze of the apartmets Perform a resdual aalyss for these data The data are stored the fle retxls Based o these results, a determe the adequacy of the ft of the model b evaluate whether the assumptos of regresso have bee serously volated 330 I Problem 38 o page 53, you used aual reveues to predct the value of a baseball frachse The data are stored the fle bbreveuexls Perform a resdual aalyss for these data Based o these results, a determe the adequacy of the ft of the model b evaluate whether the assumptos of regresso have bee serously volated 33 I Problem 30 o page 53, you used hardess to predct the tesle stregth of de-cast alumum The data are stored the fle hardessxls Perform a resdual aalyss for these data Based o these results, a determe the adequacy of the ft of the model b evaluate whether the assumptos of regresso have bee serously volated 36 MEASURING AUTOCORRELATION: THE DURBIN-WATSON STATISTIC Oe of the basc assumptos of the regresso model s the depedece of the errors Ths assumpto s sometmes volated whe data are collected over sequetal tme perods because a resdual at ay oe tme perod may ted to be smlar to resduals at adjacet tme perods Ths patter the resduals s called autocorrelato Whe a set of data has substatal autocorrelato, the valdty of a regresso model ca be serous doubt Resdual Plots to Detect Autocorrelato As metoed Secto 35, oe way to detect autocorrelato s to plot the resduals tme order If a postve autocorrelato effect s preset, there wll be clusters of resduals wth the same sg, ad you wll readly detect a apparet patter If egatve autocorrelato exsts, resduals wll ted to jump back ad forth from postve to egatve to postve, ad so o Ths type of patter s very rarely see regresso aalyss Thus, the focus of ths secto s o postve autocorrelato To llustrate postve autocorrelato, cosder the followg example The maager of a package delvery store wats to predct weekly sales, based o the umber of customers makg purchases for a perod of 5 weeks I ths stuato, because data are collected over a perod of 5 cosecutve weeks at the same store, you eed to determe whether autocorrelato s preset Table 34 presets the data (stored the fle custsalexls) Fgure 34 llustrates Mcrosoft Excel results for these data

25 36: Measurg Autocorrelato: The Durb-Watso Statstc 535 TABLE 34 Customers ad Sales for a Perod of 5 Cosecutve Weeks Sales Sales (Thousads (Thousads Week Customers of Dollars) Week Customers of Dollars) FIGURE 34 Mcrosoft Excel results for the package delvery store data of Table 34 See Secto E3 to create ths From Fgure 34, observe that r s 06574, dcatg that 6574% of the varato sales s explaed by varato the umber of customers I addto, the Y tercept, b 0, s 603, ad the slope, b, s However, before usg ths model for predcto, you must udertake proper aalyses of the resduals Because the data have bee collected over a cosecutve perod of 5 weeks, addto to checkg the learty, ormalty, ad equalvarace assumptos, you must vestgate the depedece-of-errors assumpto You ca plot the resduals versus tme to help you see whether a patter exsts I Fgure 35, you ca see that the resduals ted to fluctuate up ad dow a cyclcal patter Ths cyclcal patter provdes strog cause for cocer about the autocorrelato of the resduals ad, hece, a volato of the depedece-of-errors assumpto FIGURE 35 Mcrosoft Excel resdual plot for the package delvery store data of Table 34 See Secto E33 to create ths

26 536 CHAPTER THIRTEEN Smple Lear Regresso The Durb-Watso Statstc The Durb-Watso statstc s used to measure autocorrelato Ths statstc measures the correlato betwee each resdual ad the resdual for the tme perod mmedately precedg the oe of terest Equato (35) defes the Durb-Watso statstc DURBIN-WATSON STATISTIC D = ( e e ) e (35) where e = resdual at the tme perod To better uderstad the Durb-Watso statstc, D, you ca exame Equato (35) The umerator, ( e e ), represets the squared dfferece betwee two successve resduals, summed from the secod value to the th value The deomator, e, represets the sum of the squared resduals Whe successve resduals are postvely autocorrelated, the value of D approaches 0 If the resduals are ot correlated, the value of D wll be close to (If there s egatve autocorrelato, D wll be greater tha ad could eve approach ts maxmum value of 4) For the package delvery store data, as show the Mcrosoft Excel results of Fgure 36, the Durb-Watso statstc, D, s FIGURE 36 Mcrosoft Excel results of the Durb-Watso statstc for the package delvery store data See Secto E34 to create ths You eed to determe whe the autocorrelato s large eough to make the Durb- Watso statstc, D, fall suffcetly below to coclude that there s sgfcat postve autocorrelato After computg D, you compare t to the crtcal values of the Durb-Watso statstc foud Table E0, a porto of whch s preseted Table 35 The crtcal values deped o α, the sgfcace level chose,, the sample sze, ad k, the umber of depedet varables the model ( smple lear regresso, k = ) TABLE 35 Fdg Crtcal Values of the Durb-Watso Statstc α = 05 k = k = k = 3 k = 4 k = 5 d L D U d L d U d L d U d L d U d L d U

27 36: Measurg Autocorrelato: The Durb-Watso Statstc 537 I Table 35, two values are show for each combato of α (level of sgfcace), (sample sze), ad k (umber of depedet varables the model) The frst value, d L, represets the lower crtcal value If D s below d L, you coclude that there s evdece of postve autocorrelato amog the resduals If ths occurs, the least-squares method used ths chapter s approprate, ad you should use alteratve methods (see referece 4) The secod value, d U, represets the upper crtcal value of D, above whch you would coclude that there s o evdece of postve autocorrelato amog the resduals If D s betwee d L ad d U, you are uable to arrve at a defte cocluso For the package delvery store data, wth oe depedet varable (k = ) ad 5 values ( = 5), d L = 08 ad d U = 36 Because D = < 08, you coclude that there s postve autocorrelato amog the resduals The least-squares regresso aalyss of the data s approprate because of the presece of sgfcat postve autocorrelato amog the resduals I other words, the depedece-of-errors assumpto s vald You eed to use alteratve approaches dscussed referece 4 PROBLEMS FOR SECTION 36 Learg the Bascs PH Grade ASSIST 33 The resduals for 0 cosecutve tme perods are as follows: Tme Perod Resdual Tme Perod Resdual a Plot the resduals over tme What cocluso ca you reach about the patter of the resduals over tme? b Based o (a), what cocluso ca you reach about the autocorrelato of the resduals? PH Grade ASSIST 333 The resduals for 5 cosecutve tme perods are as follows: Tme Perod Resdual Tme Perod Resdual a Plot the resduals over tme What cocluso ca you reach about the patter of the resduals over tme? b Compute the Durb-Watso statstc At the 005 level of sgfcace, s there evdece of postve autocorrelato amog the resduals? c Based o (a) ad (b), what cocluso ca you reach about the autocorrelato of the resduals? Applyg the Cocepts PH Grade ASSIST 334 I Problem 34 o page 5 cocerg pet food sales, the marketg maager used shelf space for pet food to predct weekly sales a Is t ecessary to compute the Durb-Watso statstc ths case? Expla b Uder what crcumstaces s t ecessary to compute the Durb-Watso statstc before proceedg wth the least-squares method of regresso aalyss? 335 The ower of a sgle-famly home a suburba couty the ortheaster Uted States would lke to develop a model to predct electrcty cosumpto hs allelectrc house (lghts, fas, heat, applaces, ad so o), based o average atmospherc temperature ( degrees Fahrehet) Mothly klowatt usage ad temperature data are avalable for a perod of 4 cosecutve moths the fle elecusexls a Assumg a lear relatoshp, use the least-squares method to fd the regresso coeffcets b 0 ad b b Predct the mea klowatt usage whe the average atmospherc temperature s 50 Fahrehet c Plot the resduals versus the tme perod d Compute the Durb-Watso statstc At the 005 level of sgfcace, s there evdece of postve autocorrelato amog the resduals? e Based o the results of (c) ad (d), s there reaso to questo the valdty of the model?

28 538 CHAPTER THIRTEEN Smple Lear Regresso 336 A mal-order catalog busess that sells persoal computer supples, software, ad hardware matas a cetralzed warehouse for the dstrbuto of products ordered Maagemet s curretly examg the process of dstrbuto from the warehouse ad s terested studyg the factors that affect warehouse dstrbuto costs Curretly, a small hadlg fee s added to the order, regardless of the amout of the order Data have bee collected over the past 4 moths, dcatg the warehouse dstrbuto costs ad the umber of orders receved They are stored the fle warecostxls The results are as follows: Dstrbuto Cost Number Moths (Thousads of Dollars) of Orders 595 4, , , , , , , , , , , , , , , , , , , , , , , ,6 a Assumg a lear relatoshp, use the least-squares method to fd the regresso coeffcets b 0 ad b b Predct the mothly warehouse dstrbuto costs whe the umber of orders s 4,500 c Plot the resduals versus the tme perod d Compute the Durb-Watso statstc At the 005 level of sgfcace, s there evdece of postve autocorrelato amog the resduals? e Based o the results of (c) ad (d), s there reaso to questo the valdty of the model? 337 A freshly brewed shot of espresso has three dstct compoets: the heart, body, ad crema The separato of these three compoets typcally lasts oly 0 to 0 secods To use the espresso shot makg a latte, cappucco, or other drks, the shot must be poured to the beverage durg the separato of the heart, body, ad crema If the shot s used after the separato occurs, the drk becomes excessvely btter ad acdc, rug the fal drk Thus, a loger separato tme allows the drk-maker more tme to pour the shot ad esure that the beverage wll meet expectatos A employee at a coffee shop hypotheszed that the harder the espresso grouds were tamped dow to the portaflter before brewg, the loger the separato tme would be A expermet usg 4 observatos was coducted to test ths relatoshp The depedet varable Tamp measures the dstace, ches, betwee the espresso grouds ad the top of the portaflter (that s, the harder the tamp, the larger the dstace) The depedet varable Tme s the umber of secods the heart, body, ad crema are separated (that s, the amout of tme after the shot s poured before t must be used for the customer s beverage) The data are stored the fle espressoxls: Shot Tamp Tme Shot Tamp Tme a Determe the predcto le, usg Tme as the depedet varable ad Tamp as the depedet varable b Predct the mea separato tme for a Tamp dstace of 050 ch c Plot the resduals versus the tme order of expermetato Are there ay otceable patters? d Compute the Durb-Watso statstc At the 005 level of sgfcace, s there evdece of postve autocorrelato amog the resduals? e Based o the results of (c) ad (d), s there reaso to questo the valdty of the model? 338 The ower of a cha of ce cream stores would lke to study the effect of atmospherc temperature o sales durg the summer seaso A sample of cosecutve days s selected, wth the results stored the data fle cecreamxls (Ht: Determe whch are the depedet ad depedet varables)

29 37: Ifereces About the Slope ad Correlato Coeffcet 539 a Assumg a lear relatoshp, use the least-squares method to fd the regresso coeffcets b 0 ad b b Predct the sales per store for a day whch the temperature s 83 F c Plot the resduals versus the tme perod d Compute the Durb-Watso statstc At the 005 level of sgfcace, s there evdece of postve autocorrelato amog the resduals? e Based o the results of (c) ad (d), s there reaso to questo the valdty of the model? 37 INFERENCES ABOUT THE SLOPE AND CORRELATION COEFFICIENT I Sectos 3 through 33, regresso was used solely for descrptve purposes You leared how the least-squares method determes the regresso coeffcets ad how to predct Y for a gve value of X I addto, you leared how to compute ad terpret the stadard error of the estmate ad the coeffcet of determato Whe resdual aalyss, as dscussed Secto 35, dcates that the assumptos of a least-squares regresso model are ot serously volated ad that the straght-le model s approprate, you ca make fereces about the lear relatoshp betwee the varables the populato t Test for the Slope To determe the exstece of a sgfcat lear relatoshp betwee the X ad Y varables, you test whether β (the populato slope) s equal to 0 The ull ad alteratve hypotheses are as follows: H 0 : β = 0 (There s o lear relatoshp) H : β 0 (There s a lear relatoshp) If you reject the ull hypothess, you coclude that there s evdece of a lear relatoshp Equato (36) defes the test statstc TESTING A HYPOTHESIS FOR A POPULATION SLOPE, β, USING THE t TEST The t statstc equals the dfferece betwee the sample slope ad hypotheszed value of the populato slope dvded by the stadard error of the slope t = b β S b (36) where S b = SYX SSX SSX = ( X X) The test statstc t follows a t dstrbuto wth degrees of freedom Retur to the Usg Statstcs scearo cocerg Suflowers Apparel To test whether there s a sgfcat lear relatoshp betwee the sze of the store ad the aual sales at the 005 level of sgfcace, refer to the Mcrosoft Excel worksheet for the t test preseted Fgure 37

30 540 CHAPTER THIRTEEN Smple Lear Regresso FIGURE 37 Mcrosoft Excel t test for the slope for the Suflowers Apparel data From Fgure 37, See Secto E3 to create the worksheet that cotas ths area ad b = = 4 S b = t b β = S b = = 0 64 Mcrosoft Excel labels ths t statstc t Stat (see Fgure 37) Usg the 005 level of sgfcace, the crtcal value of t wth = degrees of freedom s 788 Because t = 064 > 788, you reject H 0 (see Fgure 38) Usg the p-value, you reject H 0 because the p-value s approxmately 0 whch s less tha α = 005 Hece, you ca coclude that there s a sgfcat lear relatoshp betwee mea aual sales ad the sze of the store FIGURE 38 Testg a hypothess about the populato slope at the 005 level of sgfcace, wth degrees of freedom t Rego of Rejecto Rego of Norejecto Rego of Rejecto Crtcal Value Crtcal Value F Test for the Slope As a alteratve to the t test, you ca use a F test to determe whether the slope smple lear regresso s statstcally sgfcat I Secto 04, you used the F dstrbuto to test the rato of two varaces Equato (37) defes the F test for the slope as the rato of the varace that s due to the regresso (MSR) dvded by the error varace (MSE = ) S YX TESTING A HYPOTHESIS FOR A POPULATION SLOPE, β, USING THE F TEST The F statstc s equal to the regresso mea square (MSR) dvded by the error mea square (MSE) F = MSR MSE (37)

31 37: Ifereces About the Slope ad Correlato Coeffcet 54 where SSR MSR = k SSE MSE = k k = umber of depedet varables the regresso model The test statstc F follows a F dstrbuto wth k ad k degrees of freedom Usg a level of sgfcace α, the decso rule s Reject H 0 f F > F U ; otherwse, do ot reject H 0 Table 36 orgazes the complete set of results to a ANOVA table TABLE 36 ANOVA Table for Testg the Sgfcace of a Regresso Coeffcet Sum of Mea Square Source df Squares (Varace) F Regresso k SSR Error k SSE Total SST MSR = MSE SSR k SSE = k F = MSR MSE The completed ANOVA table s also part of the Mcrosoft Excel results show Fgure 39 Fgure 39 shows that the computed F statstc s 3335 ad the p-value s approxmately 0 FIGURE 39 Mcrosoft Excel F test for the Suflowers Apparel data See Secto E3 to create the worksheet that cotas ths area Usg a level of sgfcace of 005, from Table E5, the crtcal value of the F dstrbuto, wth ad degrees of freedom, s 475 (see Fgure 30) Because F = 3335 > 475 or because the p-value = < 005, you reject H 0 ad coclude that the sze of the store s sgfcatly related to aual sales Because the F test Equato 37 o page 540 s equvalet to the t test o page 539, you reach the same cocluso

32 54 CHAPTER THIRTEEN Smple Lear Regresso FIGURE 30 Regos of rejecto ad orejecto whe testg for sgfcace of slope at the 005 level of sgfcace, wth ad degrees of freedom F Rego of Norejecto Crtcal Value Rego of Rejecto Cofdece Iterval Estmate of the Slope (β ) As a alteratve to testg for the exstece of a lear relatoshp betwee the varables, you ca costruct a cofdece terval estmate of β ad determe whether the hypotheszed value (β = 0) s cluded the terval Equato (38) defes the cofdece terval estmate of β CONFIDENCE INTERVAL ESTIMATE OF THE SLOPE, β The cofdece terval estmate for the slope ca be costructed by takg the sample slope, b, ad addg ad subtractg the crtcal t value multpled by the stadard error of the slope b t S ± b (38) From the Mcrosoft Excel results of Fgure 37 o page 540, b = 6699 = 4 S b = To costruct a 95% cofdece terval estmate, α/ = 005, ad from Table E3, t = 788 Thus, b ± t Sb = 6699 ± ( 788)( 0 569) = 6699 ± β 08 Therefore, you estmate wth 95% cofdece that the populato slope s betwee 380 ad 08 Because these values are above 0, you coclude that there s a sgfcat lear relatoshp betwee aual sales ad the sze of the store Had the terval cluded 0, you would have cocluded that o sgfcat relatoshp exsts betwee the varables The cofdece terval dcates that for each crease of,000 square feet, mea aual sales are estmated to crease by at least $,38,000 but o more tha $,0,800 t Test for the Correlato Coeffcet I Secto 35 o page 30, the stregth of the relatoshp betwee two umercal varables was measured, usg the correlato coeffcet, r You ca use the correlato coeffcet to determe whether there s a statstcally sgfcat lear relatoshp betwee X ad Y To do

33 37: Ifereces About the Slope ad Correlato Coeffcet 543 so, you hypothesze that the populato correlato coeffcet, ρ, s 0 Thus, the ull ad alteratve hypotheses are H 0 : ρ = 0 (o correlato) H : ρ 0 (correlato) Equato (39) defes the test statstc for determg the exstece of a sgfcat correlato TESTING FOR THE EXISTENCE OF CORRELATION r ρ t = r where r = + r f b > 0 r = r f b < 0 (39) The test statstc t follows a t dstrbuto wth degrees of freedom I the Suflowers Apparel problem, r = 0904 ad b = (see Fgure 34 o page 56) Because b > 0, the correlato coeffcet for aual sales ad store sze s the postve square root of r, that s, r = = Testg the ull hypothess that there s o correlato betwee these two varables results the followg observed t statstc: t = = r 0 r ( ) 4 = 0 64 Usg the 005 level of sgfcace, because t = 064 > 788, you reject the ull hypothess You coclude that there s evdece of a assocato betwee aual sales ad store sze Ths t statstc s equvalet to the t statstc foud whe testg whether the populato slope, β, s equal to zero (see Fgure 37 o page 540) Whe fereces cocerg the populato slope were dscussed, cofdece tervals ad tests of hypothess were used terchageably However, developg a cofdece terval for the correlato coeffcet s more complcated because the shape of the samplg dstrbuto of the statstc r vares for dfferet values of the populato correlato coeffcet Methods for developg a cofdece terval estmate for the correlato coeffcet are preseted referece 4 PROBLEMS FOR SECTION 37 Learg the Bascs 339 You are testg the ull hypothess that there s o lear relatoshp betwee two varables, X ad Y From your sample of = 0, you determe that r = 080 a What s the value of the t test statstc? b At the α = 005 level of sgfcace, what are the crtcal values? c Based o your aswers to (a) ad (b), what statstcal decso should you make?

34 544 CHAPTER THIRTEEN Smple Lear Regresso PH Grade ASSIST 340 You are testg the ull hypothess that there s o relatoshp betwee two varables, X ad Y From your sample of = 8, you determe that b = +45 ad S b = 5 a What s the value of the t test statstc? b At the α = 005 level of sgfcace, what are the crtcal values? c Based o your aswers to (a) ad (b), what statstcal decso should you make? d Costruct a 95% cofdece terval estmate of the populato slope, β PH Grade ASSIST 34 You are testg the ull hypothess that there s o relatoshp betwee two varables, X ad Y From your sample of = 0, you determe that SSR =60adSSE = 40 a What s the value of the F test statstc? b At the α = 005 level of sgfcace, what s the crtcal value? c Based o your aswers to (a) ad (b), what statstcal decso should you make? d Compute the correlato coeffcet by frst computg r ad assumg that b s egatve e At the 005 level of sgfcace, s there a sgfcat correlato betwee X ad Y? Applyg the Cocepts SELF 34 I Problem 34 o page 5, the marketg maager used shelf space for pet food to pre- Test dct weekly sales The data are stored the fle petfoodxls From the results of that problem, b = 74 ad S b = 59 a At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee shelf space ad sales? b Costruct a 95% cofdece terval estmate of the populato slope, β 343 I Problem 35 o page 5, you used reported magaze ewsstad sales to predct audted sales The data are stored the fle crculatoxls Usg the results of that problem, b = 0579 ad S b = a At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee reported sales ad audted sales? b Costruct a 95% cofdece terval estmate of the populato slope, β 344 I Problem 36 o pages 5 53, the ower of a movg compay wated to predct labor hours, based o the umber of cubc feet moved The data are stored the fle movgxls Usg the results of that problem, a at the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the umber of cubc feet moved ad labor hours? b costruct a 95% cofdece terval estmate of the populato slope, β PH Grade ASSIST 345 I Problem 37 o page 53, you used the weght of mal to predct the umber of orders receved The data are stored the fle malxls Usg the results of that problem, a at the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the weght of mal ad the umber of orders receved? b costruct a 95% cofdece terval estmate of the populato slope, β 346 I Problem 38 o page 53, you used aual reveues to predct the value of a baseball frachse The data are stored the fle bbreveuexls Usg the results of that problem, a at the 005 level of sgfcace, s there evdece of a lear relatoshp betwee aual reveue ad frachse value? b costruct a 95% cofdece terval estmate of the populato slope, β 347 I Problem 39 o page 53, a aget for a real estate compay wated to predct the mothly ret for apartmets, based o the sze of the apartmet The data are stored the fle retxls Usg the results of that problem, a at the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the sze of the apartmet ad the mothly ret? b costruct a 95% cofdece terval estmate of the populato slope, β 348 I Problem 30 o page 53, you used hardess to predct the tesle stregth of de-cast alumum The data are stored the fle hardessxls Usg the results of that problem, a at the 005 level of sgfcace, s there evdece of a lear relatoshp betwee hardess ad tesle stregth? b costruct a 95% cofdece terval estmate of the populato slope, β 349 The volatlty of a stock s ofte measured by ts beta value You ca estmate the beta value of a stock by developg a smple lear regresso model, usg the percetage weekly chage the stock as the depedet varable ad the percetage weekly chage a market dex as the depedet varable The S&P 500 Idex s a commo dex to use For example, f you wated to estmate the beta for IBM, you could use the followg model, whch s sometmes referred to as a market model: (% weekly chage IBM) = β 0 + β (% weekly chage S & P 500 dex) + ε The least-squares regresso estmate of the slope b s the estmate of the beta value for IBM A stock wth a beta value of 0 teds to move the same as the overall market A stock wth a beta value of 5 teds to move 50% more tha the overall market, ad a stock wth a beta value of 06

35 37: Ifereces About the Slope ad Correlato Coeffcet 545 teds to move oly 60% as much as the overall market Stocks wth egatve beta values ted to move a drecto opposte that of the overall market The followg table gves some beta values for some wdely held stocks: Compay Tcker Symbol Beta AT&T T 080 IBM IBM 0 Dsey Compay DIS 40 Alcoa AA 6 LSI Logc LSI 36 Source: Extracted from faceyahoocom, May 3, 006 a For each of the fve compaes, terpret the beta value b How ca vestors use the beta value as a gude for vestg? 350 Idex fuds are mutual fuds that try to mmc the movemet of leadg dexes, such as the S&P 500 Idex, the NASDAQ 00 Idex, or the Russell 000 Idex The beta values for these fuds (as descrbed Problem 349) are therefore approxmately 0 The estmated market models for these fuds are approxmately (% weekly chage dex fud) = (% weekly chage the dex) Leveraged dex fuds are desged to magfy the movemet of major dexes A artcle Mutual Fuds (L O Shaughessy, Reach for Hgher Returs, Mutual Fuds, July 999, pp 44 49) descrbed some of the rsks ad rewards assocated wth these fuds ad gave detals o some of the most popular leveraged fuds, cludg those the followg table: Name (Tcker Symbol) Fud Descrpto Potomac Small Cap 5% of Russell 000 Idex Plus (POSCX) Rydex Iv Nova (RYNVX) 50% of the S&P 500 Idex ProFud UltraOTC Double (00%) the NASDAQ 00 Iv (UOPIX) Idex Thus, estmated market models for these fuds are approxmately (% weekly chage POSCX) = (% weekly chage the Russell 000 Idex) (% weekly chage RYNVX) = (% weekly chage the S&P 500 Idex) (% weekly chage UOPIX fud) = (% weekly chage the NASDAQ 00 Idex) Thus, f the Russell 000 Idex gas 0% over a perod of tme, the leveraged mutual fud POSCX gas approxmately 5% O the dowsde, f the same dex loses 0%, POSCX loses approxmately 5% a Cosder the leveraged mutual fud ProFud UltraOTC Iv (UOPIX), whose descrpto s 00% of the performace of the S&P 500 Idex What s ts approxmate market model? b If the NASDAQ gas 30% a year, what retur do you expect UOPIX to have? c If the NASDAQ loses 35% a year, what retur do you expect UOPIX to have? d What type of vestors should be attracted to leveraged fuds? What type of vestors should stay away from these fuds? 35 The data the fle coffeedrkxls represet the calores ad fat ( grams) of 6-ouce ced coffee drks at Duk Douts ad Starbucks: Product Calores Fat Duk Douts Iced Mocha Swrl latte (whole mlk) Starbucks Coffee Frappucco bleded coffee Duk Douts Coffee Coolatta (cream) Starbucks Iced Coffee Mocha Espresso (whole mlk ad whpped cream) Starbucks Mocha Frappucco bleded coffee (whpped cream) Starbucks Chocolate Browe Frappucco bleded coffee (whpped cream) 50 0 Starbucks Chocolate Frappucco Bleded Crème (whpped cream) Source: Extracted from Coffee as Cady at Duk Douts ad Starbucks, Cosumer Reports, Jue 004, p 9 a Compute ad terpret the coeffcet of correlato, r b At the 005 level of sgfcace, s there a sgfcat lear relatoshp betwee the calores ad fat? 35 There are several methods for calculatg fuel ecoomy The followg table (cotaed the fle mleagexls) dcates the mleage as calculated by owers ad by curret govermet stadards: Govermet Vehcle Ower Stadards 005 Ford F Chevrolet Slverado Hoda Accord LX Hoda Cvc Hoda Cvc Hybrd Ford Explorer Toyota Camry Toyota Corolla Toyota Prus

36 546 CHAPTER THIRTEEN Smple Lear Regresso a Compute ad terpret the coeffcet of correlato, r b At the 005 level of sgfcace, s there a sgfcat lear relatoshp betwee the mleage as calculated by owers ad by curret govermet stadards? 353 College basketball s bg busess, wth coaches salares, reveues, ad expeses mllos of dollars The data the fle colleges-basketballxls represet the coaches salares ad reveues for college basketball at selected schools a recet year (extracted from R Adams, Pay for Playoffs, The Wall Street Joural, March, 006, pp P, P8) a Compute ad terpret the coeffcet of correlato, r b At the 005 level of sgfcace, s there a sgfcat lear relatoshp betwee a coach s salary ad reveue? 354 College football players tryg out for the NFL are gve the Woderlc stadardzed tellgece test The data the fle woderlcxls represet the average Woderlc scores of football players tryg out for the NFL ad the graduato rates for football players at selected schools (extracted from S Walker, The NFL s Smartest Team, The Wall Street Joural, September 30, 005, pp W, W0) a Compute ad terpret the coeffcet of correlato, r b At the 005 level of sgfcace, s there a sgfcat lear relatoshp betwee the average Woderlc score of football players tryg out for the NFL ad the graduato rates for football players at selected schools? c What coclusos ca you reach about the relatoshp betwee the average Woderlc score of football players tryg out for the NFL ad the graduato rates for football players at selected schools? 38 ESTIMATION OF MEAN VALUES AND PREDICTION OF INDIVIDUAL VALUES Ths secto presets methods of makg fereces about the mea of Y ad predctg dvdual values of Y The Cofdece Iterval Estmate I Example 3 o page 59, you used the predcto le to predct the value of Y for a gve X The mea aual sales for stores wth 4,000 square feet was predcted to be 7644 mllos of dollars ($7,644,000) Ths estmate, however, s a pot estmate of the populato mea I Chapter 8, you studed the cocept of the cofdece terval as a estmate of the populato mea I a smlar fasho, Equato (30) defes the cofdece terval estmate for the mea respose for a gve X CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN OF Y Yˆ ± t SYX h Yˆ t S h µ Yˆ + t S h where h YX Y X = X YX ( X X) = + SSX Yˆ = predcted value of Y; Yˆ = b0 + bx S YX = stadard error of the estmate = sample sze X = gve value of X (30) µ YX = X = mea value of Y whe X = X SSX = ( X X)

37 38: Estmato of Mea Values ad Predcto of Idvdual Values 547 The wdth of the cofdece terval Equato (30) depeds o several factors For a gve level of cofdece, creased varato aroud the predcto le, as measured by the stadard error of the estmate, results a wder terval However, as you would expect, creased sample sze reduces the wdth of the terval I addto, the wdth of the terval also vares at dfferet values of X Whe you predct Y for values of X close to X, the terval s arrower tha for predctos for X values more dstat from X I the Suflowers Apparel example, suppose you wat to costruct a 95% cofdece terval estmate of the mea aual sales for the etre populato of stores that cota 4,000 square feet (X = 4) Usg the smple lear regresso equato, Also, gve the followg: From Table E3, t = 788 Thus, where Yˆ = X = ( 4) = (mllos of dollars) X = 94 S = YX SSX = ( X X ) = Yˆ t S h ± YX h ( X X) = + SSX so that ˆ ( X X) Y ± t SYX + SSX ( 4 94) = ± ( 788)( ) = ± so 697 µ Y X = Therefore, the 95% cofdece terval estmate s that the mea aual sales are betwee $6,97,00 ad $8,36,700 for the populato of stores wth 4,000 square feet The Predcto Iterval I addto to the eed for a cofdece terval estmate for the mea value, you ofte wat to predct the respose for a dvdual value Although the form of the predcto terval s smlar to that of the cofdece terval estmate of Equato (30), the predcto terval s predctg a dvdual value, ot estmatg a parameter Equato (3) defes the predcto terval for a dvdual respose, Y, at a partcular value, X, deoted by Y X = X

38 548 CHAPTER THIRTEEN Smple Lear Regresso PREDICTION INTERVAL FOR AN INDIVIDUAL RESPONSE, Y Yˆ ± t S + h YX Yˆ t S + h Y Yˆ + t S + h YX X= X YX (3) where h, Yˆ, S YX,, ad X are defed as Equato (30) o page 546 ad Y X = X s a future value of Y whe X = X To costruct a 95% predcto terval of the aual sales for a dvdual store that cotas 4,000 square feet (X = 4), you frst compute ˆ Usg the predcto le: = (mllos of dollars) Also, gve the followg: X = 94 S = From Table E3, t = 788 Thus, where Yˆ = X = ( 4) YX SSX = ( X X ) = Y Yˆ ± t S + h YX h = + ( X X) ( X X) so that ˆ ( X X) Y ± t SYX + + SSX ( 4 94) = ± ( 788)( ) = ± 04 so Y X= Therefore, wth 95% cofdece, you predct that the aual sales for a dvdual store wth 4,000 square feet s betwee $5,433,500 ad $9,854,300

39 38: Estmato of Mea Values ad Predcto of Idvdual Values 549 Fgure 3 s a Mcrosoft Excel worksheet that llustrates the cofdece terval estmate ad the predcto terval for the Suflowers Apparel problem If you compare the results of the cofdece terval estmate ad the predcto terval, you see that the wdth of the predcto terval for a dvdual store s much wder tha the cofdece terval estmate for the mea Remember that there s much more varato predctg a dvdual value tha estmatg a mea value FIGURE 3 Mcrosoft Excel cofdece terval estmate ad predcto terval for the Suflowers Apparel data See Secto E35 to create ths PROBLEMS FOR SECTION 38 Learg the Bascs PH Grade ASSIST 355 Based o a sample of = 0, the leastsquares method was used to develop the followg predcto le: ˆ = 5 + 3X I addto, a Costruct a 95% cofdece terval estmate of the populato mea respose for X = b Costruct a 95% predcto terval of a dvdual respose for X = PH Grade ASSIST S = 0 X = ( X X) = 0 YX 356 Based o a sample of = 0, the leastsquares method was used to develop the followg predcto le: ˆ = 5 + 3X I addto, S = 0 X = ( X X) = 0 YX a Costruct a 95% cofdece terval estmate of the populato mea respose for X = 4 b Costruct a 95% predcto terval of a dvdual respose for X = 4 c Compare the results of (a) ad (b) wth those of Problem 355 (a) ad (b) Whch terval s wder? Why? Y Y Applyg the Cocepts 357 I Problem 35 o page 5, you used reported sales to predct audted sales of magazes The data are stored the fle crculatoxls For these data S YX = 486 ad h = 008 whe X = 400 a Costruct a 95% cofdece terval estmate of the mea audted sales for magazes that report ewsstad sales of 400,000 b Costruct a 95% predcto terval of the audted sales for a dvdual magaze that reports ewsstad sales of 400,000 c Expla the dfferece the results (a) ad (b) PH Grade ASSIST SELF Test 358 I Problem 34 o page 5, the marketg maager used shelf space for pet food to predct weekly sales The data are stored the fle petfoodxls For these data S YX = 308 ad h = 0373 whe X = 8 a Costruct a 95% cofdece terval estmate of the mea weekly sales for all stores that have 8 feet of shelf space for pet food b Costruct a 95% predcto terval of the weekly sales of a dvdual store that has 8 feet of shelf space for pet food c Expla the dfferece the results (a) ad (b)

40 550 CHAPTER THIRTEEN Smple Lear Regresso 359 I Problem 37 o page 53, you used the weght of mal to predct the umber of orders receved The data are stored the fle malxls a Costruct a 95% cofdece terval estmate of the mea umber of orders receved for all packages wth a weght of 500 pouds b Costruct a 95% predcto terval of the umber of orders receved for a dvdual package wth a weght of 500 pouds c Expla the dfferece the results (a) ad (b) 360 I Problem 36 o page 5, the ower of a movg compay wated to predct labor hours based o the umber of cubc feet moved The data are stored the fle movgxls a Costruct a 95% cofdece terval estmate of the mea labor hours for all moves of 500 cubc feet b Costruct a 95% predcto terval of the labor hours of a dvdual move that has 500 cubc feet c Expla the dfferece the results (a) ad (b) 36 I Problem 39 o page 53, a aget for a real estate compay wated to predct the mothly ret for apartmets, based o the sze of the apartmet The data are stored the fle retxls a Costruct a 95% cofdece terval estmate of the mea mothly retal for all apartmets that are,000 square feet sze b Costruct a 95% predcto terval of the mothly retal of a dvdual apartmet that s,000 square feet sze c Expla the dfferece the results (a) ad (b) 36 I Problem 38 o page 53, you predcted the value of a baseball frachse, based o curret reveue The data are stored the fle bbreveuexls a Costruct a 95% cofdece terval estmate of the mea value of all baseball frachses that geerate $50 mllo of aual reveue b Costruct a 95% predcto terval of the value of a dvdual baseball frachse that geerates $50 mllo of aual reveue c Expla the dfferece the results (a) ad (b) 363 I Problem 30 o page 53, you used hardess to predct the tesle stregth of de-cast alumum The data are stored the fle hardessxls a Costruct a 95% cofdece terval estmate of the mea tesle stregth for all specmes wth a hardess of 30 Rockwell E uts b Costruct a 95% predcto terval of the tesle stregth for a dvdual specme that has a hardess of 30 Rockwell E uts c Expla the dfferece the results (a) ad (b) 39 PITFALLS IN REGRESSION AND ETHICAL ISSUES Some of the ptfalls volved usg regresso aalyss are as follows: Lackg a awareess of the assumptos of least-squares regresso Not kowg how to evaluate the assumptos of least-squares regresso Not kowg what the alteratves to least-squares regresso are f a partcular assumpto s volated Usg a regresso model wthout kowledge of the subject matter Extrapolatg outsde the relevat rage Cocludg that a sgfcat relatoshp detfed a observatoal study s due to a cause-ad-effect relatoshp The wdespread avalablty of spreadsheet ad statstcal software has made regresso aalyss much more feasble However, for may users, ths ehaced avalablty of software has ot bee accompaed by a uderstadg of how to use regresso aalyss properly Someoe who s ot famlar wth ether the assumptos of regresso or how to evaluate the assumptos caot be expected to kow what the alteratves to least-squares regresso are f a partcular assumpto s volated The data Table 37 (stored the fle ascombexls) llustrate the mportace of usg scatter plots ad resdual aalyss to go beyod the basc umber cruchg of computg the Y tercept, the slope, ad r

41 39: Ptfalls Regresso ad Ethcal Issues 55 TABLE 37 Four Sets of Artfcal Data Data Set A Data Set B Data Set C Data Set D X Y X Y X Y X Y Source: Extracted from F J Ascombe, Graphs Statstcal Aalyss, Amerca Statstca, Vol 7 (973), pp 7 Ascombe (referece ) showed that all four data sets gve Table 37 have the followg detcal results: Yˆ = X SYX = 37 S = 0 8 b r = SSR = Explaed varato = ( Yˆ Y ) = 7 5 SSE = Uxplaed varato = ( Y Yˆ ) = 3 76 SST = Total varato = ( Y Y ) = 4 7 Thus, wth respect to these statstcs assocated wth a smple lear regresso aalyss, the four data sets are detcal Were you to stop the aalyss at ths pot, you would fal to observe the mportat dffereces amog the four data sets By examg the scatter plots for the four data sets Fgure 3 o page 55, ad ther resdual plots Fgure 33 o page 55, you ca clearly see that each of the four data sets has a dfferet relatoshp betwee X ad Y From the scatter plots of Fgure 3 ad the resdual plots of Fgure 33, you see how dfferet the data sets are The oly data set that seems to follow a approxmate straght le s data set A The resdual plot for data set A does ot show ay obvous patters or outlyg resduals Ths s certaly ot true for data sets B, C, ad D The scatter plot for data set B shows that a quadratc regresso model (see Secto 5) s more approprate Ths cocluso s reforced by the resdual plot for data set B The scatter plot ad the resdual plot for data set C clearly show a outlyg observato If ths s the case, you may wat to remove the outler ad reestmate the regresso model (see referece 4) Smlarly, the scatter plot for data set D represets the stuato whch the model s heavly depedet o the outcome of a sgle respose (X 8 = 9 ad Y 8 = 50) You would have to cautously evaluate ay regresso model because ts regresso coeffcets are heavly depedet o a sgle observato

42 55 CHAPTER THIRTEEN Smple Lear Regresso FIGURE 3 Scatter plots for four data sets Y 0 Y Pael A Pael B Y Y Pael C 5 0 Pael D 5 0 FIGURE 33 Resdual plots for four data sets Resdual + Resdual X Pael A X Pael B Resdual +4 Resdual X Pael C X Pael D

43 39: Ptfalls Regresso ad Ethcal Issues 553 I summary, scatter plots ad resdual plots are of vtal mportace to a complete regresso aalyss The formato they provde s so basc to a credble aalyss that you should always clude these graphcal methods as part of a regresso aalyss Thus, a strategy that you ca use to help avod the ptfalls of regresso s as follows: Start wth a scatter plot to observe the possble relatoshp betwee X ad Y Check the assumptos of regresso before movg o to usg the results of the model 3 Plot the resduals versus the depedet varable to determe whether the lear model s approprate ad to check the equal-varace assumpto 4 Use a hstogram, stem-ad-leaf dsplay, box-ad-whsker plot, or ormal probablty plot of the resduals to check the ormalty assumpto 5 If you collected the data over tme, plot the resduals versus tme ad use the Durb- Watso test to check the depedece assumpto 6 If there are volatos of the assumptos, use alteratve methods to least-squares regresso or alteratve least-squares models 7 If there are o volatos of the assumptos, carry out tests for the sgfcace of the regresso coeffcets ad develop cofdece ad predcto tervals 8 Avod makg predctos ad forecasts outsde the relevat rage of the depedet varable 9 Keep md that the relatoshps detfed observatoal studes may or may ot be due to cause-ad-effect relatoshps Remember that whle causato mples correlato, correlato does ot mply causato Amerca s Top Models From the Author s Desktop Perhaps you are famlar wth the TV competto orgazed by model Tyra Baks to fd Amerca s top model You may be less famlar wth aother set of top models that are emergg from the busess world I a Busess Week artcle from ts Jauary 3, 006, edto (S Baker, Why Math Wll Rock Your World: More Math Geeks Are Callg the Shots Busess Is Your Idustry Next? Busess Week, pp 54 6), Stephe Baker talks about how quats tured face upsde dow ad s movg o to other busess felds The ame quats derves from the fact that math geeks develop models ad forecasts by usg quattatve methods These methods are bult o the prcples of regresso aalyss dscussed ths chapter, although the actual models are much more complcated tha the smple lear models dscussed ths chapter Regresso-based models have become the top models for may types of busess aalyses Some examples clude Advertsg ad marketg Maagers use ecoometrc models ( other words, regresso models) to determe the effect of a advertsemet o sales, based o a set of factors Also, maagers use data mg to predct patters of behavor of what customers wll buy the future, based o hstorc formato about the cosumer Face Ay tme you read about a facal model, you should uderstad that some type of regresso model s beg used For example, a New York Tmes artcle o Jue 8, 006, ttled A Old Formula That Pots to New Worry by Mark Hulbert (p BU8) dscusses a market tmg model that predcts the retur of stocks the ext three to fve years, based o the dvded yeld of the stock market ad the terest rate of 90-day Treasury blls Food ad beverage Beleve t or ot, Eologx, a Calfora cosultg compay, has developed a formula (a regresso model) that predcts a we s qualty dex, based o a set of chemcal compouds foud the we (see D Darlgto, The Chemstry of a 90+ We, The New York Tmes Magaze, August 7, 005, pp 36 39) Publshg A study of the effect of prce chages at Amazocom ad BNcom o sales (aga, regresso aalyss) foud that a % prce chage at BNcom pushed sales dow 4%, but t pushed sales dow oly 05% at Amazocom (You ca dowload the paper at uchcagoedu/vtaehtm) Trasportato Farecastcom uses data mg ad predctve techologes to objectvely predct arfare prcg (see D Darl, Arfares Made Easy (Or Easer), The New York Tmes, July, 006, pp C, C6) Real estate Zllowcom uses formato about the features cotaed a home ad ts locato to develop estmates about the market value of the home, usg a formula bult wth a propretary algorthm I the artcle, Baker stated that statstcs ad probablty wll become core sklls for busesspeople ad cosumers Those who are successful wll kow how to use statstcs, whether they are buldg facal models or makg marketg plas He also strogly edorsed the eed for everyoe busess to have kowledge of Mcrosoft Excel to be able to produce statstcal aalyss ad reports

44 554 CHAPTER THIRTEEN Smple Lear Regresso SUMMARY As you ca see from the chapter roadmap Fgure 34, ths chapter develops the smple lear regresso model ad dscusses the assumptos ad how to evaluate them Oce you are assured that the model s approprate, you ca predct values by usg the predcto le ad test for the sgfcace of the slope Smple Lear Regresso ad Correlato Regresso Least-Squares Regresso Aalyss Scatter Plot Prmary Focus Correlato Coeffcet of Correlato, r Testg H 0 : ρ = 0 Predcto Le Plot Resduals over Tme Compute Durb-Watso Statstc Yes Data Collected Sequetal Order? No Resdual Aalyss Use Alteratve to Least-Squares Regresso Yes Is Autocorrelato Preset? No Yes Model Approprate? No Testg H 0 : β = 0 (See Assumptos) No Model Sgfcat? Yes Use Model for Predcto ad Estmato Estmate β Estmate µ YlX=X Predct Y X=X FIGURE 34 Roadmap for smple lear regresso

45 Key Equatos 555 You have leared how the drector of plag for a cha of clothg stores ca use regresso aalyss to vestgate the relatoshp betwee the sze of a store ad ts aual sales You have used ths aalyss to make better decsos whe selectg ew stes for stores as well as to forecast sales for exstg stores I Chapter 4, regresso aalyss s exteded to stuatos whch more tha oe depedet varable s used to predct the value of a depedet varable KEY EQUATIONS Smple Lear Regresso Model Y = β 0 + β X + ε (3) Smple Lear Regresso Equato: The Predcto Le Yˆ = b + b X (3) Computatoal Formula for the Slope, b (33) Computatoal Formula for the Y Itercept, b 0 b0 = Y bx (34) Measures of Varato Regresso Total Sum of Squares (SST ) SST = SSR + SSE (35) SST = Total sum of squares = ( Y Y ) Regresso Sum of Squares (SSR) SSR = Explaed varato or regresso of squares = ( Yˆ Y ) Error Sum of Squares (SSE) SSE = Uexplaed varato or error sum of squares = ( Y Yˆ ) Coeffcet of Determato r Regresso sum of squares = = Total sum of squares Computatoal Formula for SST Y SST = ( Y Y ) = Y b = 0 SSXY SSX SSR SST (36) (37) (38) (39) (30) Computatoal Formula for SSR Computatoal Formula for SSE Stadard Error of the Estmate Resdual SSE = ( Y Yˆ) = Y b0 Y b X Y S YX = Durb-Watso Statstc SSR = ( Yˆ Y ) = b Y + b X Y ( Y Yˆ ) SSE = D = e = Y Yˆ Testg a Hypothess for a Populato Slope, β, Usg the t Test t Testg a Hypothess for a Populato Slope, β, Usg the F Test F = ( e e ) = b 0 β S b MSR MSE (3) (3) (33) (34) (35) (36) (37) Cofdece Iterval Estmate of the Slope, β b t S (38) ± b b t S β b + t S b b e Y

46 556 CHAPTER THIRTEEN Smple Lear Regresso Testg for the Exstece of Correlato Cofdece Iterval Estmate for the Mea of Y Yˆ ± t S h YX t = r ρ r Yˆ t S h µ Yˆ + t S h YX Y X = X YX (39) (30) Predcto Iterval for a Idvdual Respose, Y Yˆ ± t S + h YX Yˆ t S + h Y Yˆ + t S + h YX X= X YX (3) KEY TERMS assumptos of regresso 59 autocorrelato 534 coeffcet of determato 56 cofdece terval estmate for the mea respose 546 correlato coeffcet 54 depedet varable 5 Durb-Watso statstc 536 error sum of squares (SSE) 54 equal varace 530 explaed varato 54 explaatory varable 53 homoscedastcty 530 depedece of errors 59 depedet varable 5 least-squares method 56 lear relatoshp 5 ormalty 530 predcto terval for a dvdual respose, Y 547 predcto le 55 regresso aalyss 5 regresso coeffcet 56 regresso sum of squares (SSR) 54 relevat rage 59 resdual 530 resdual aalyss 530 respose varable 53 scatter dagram 5 scatter plot 5 smple lear regresso 5 smple lear regresso equato 55 slope 53 stadard error of the estmate 58 total sum of squares (SST) 54 total varato 54 uexplaed varato 54 Y tercept 53 CHAPTER REVIEW PROBLEMS Checkg Your Uderstadg 364 What s the terpretato of the Y tercept ad the slope the smple lear regresso equato? 365 What s the terpretato of the coeffcet of determato? 366 Whe s the uexplaed varato (that s, error sum of squares) equal to 0? 367 Whe s the explaed varato (that s, regresso sum of squares) equal to 0? 368 Why should you always carry out a resdual aalyss as part of a regresso model? 369 What are the assumptos of regresso aalyss? 370 How do you evaluate the assumptos of regresso aalyss? 37 Whe ad how do you use the Durb-Watso statstc? 37 What s the dfferece betwee a cofdece terval estmate of the mea respose, µ YX = X, ad a predcto terval of Y X = X? Applyg the Cocepts 373 Researchers from the Lub School of Busess at Pace Uversty New York Cty coducted a study o Iteret-supported courses I oe part of the study, four umercal varables were collected o 08 studets a troductory maagemet course that met oce a week for a etre semester Oe varable collected was ht cosstecy To measure ht cosstecy, the researchers dd the followg: If a studet dd ot vst the Iteret ste betwee classes, the studet was gve a 0 for that tme perod If a studet vsted the Iteret ste oe or more tmes betwee classes, the studet was gve a for that tme perod Because there were 3 tme perods, a studet s score o ht cosstecy could rage from 0 to 3 The other three varables cluded the studet s course average, the studet s cumulatve grade pot average

47 Chapter Revew Problems 557 (GPA), ad the total umber of hts the studet had o the Iteret ste supportg the course The followg table gves the correlato coeffcet for all pars of varables Note that correlatos marked wth a * are statstcally sgfcat, usg α = 000: Varable Correlato Course Average, Cumulatve GPA 07* Course Average, Total Hts 008 Course Average, Ht Cosstecy 037* Cumulatve GPA, Total Hts 0 Cumulatve GPA, Ht Cosstecy 03* Total Hts, Ht Cosstecy 064* Source: Extracted from D Baugher, A Varaell, ad E Wesbord, Studet Hts a Iteret-Supported Course: How Ca Istructors Use Them ad What Do They Mea? Decso Sceces Joural of Iovatve Educato, Fall 003, (), pp a What coclusos ca you reach from ths correlato aalyss? b Are you surprsed by the results, or are they cosstet wth your ow observatos ad expereces? 374 Maagemet of a soft-drk bottlg compay wats to develop a method for allocatg delvery costs to customers Although oe cost clearly relates to travel tme wth a partcular route, aother varable cost reflects the tme requred to uload the cases of soft drk at the delvery pot A sample of 0 delveres wth a terrtory was selected The delvery tmes ad the umbers of cases delvered were recorded the delveryxls fle: Delvery Delvery Number Tme Number Tme Customer of Cases (Mutes) Customer of Cases (Mutes) Develop a regresso model to predct delvery tme, based o the umber of cases delvered a Use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of b 0 ad b ths problem c Predct the delvery tme for 50 cases of soft drk d Should you use the model to predct the delvery tme for a customer who s recevg 500 cases of soft drk? Why or why ot? e Determe the coeffcet of determato, r, ad expla ts meag ths problem f Perform a resdual aalyss Is there ay evdece of a patter the resduals? Expla g At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee delvery tme ad the umber of cases delvered? h Costruct a 95% cofdece terval estmate of the mea delvery tme for 50 cases of soft drk Costruct a 95% predcto terval of the delvery tme for a sgle delvery of 50 cases of soft drk j Costruct a 95% cofdece terval estmate of the populato slope k Expla how the results (a) through ( j) ca help allocate delvery costs to customers 375 A brokerage house wats to predct the umber of trade executos per day, usg the umber of comg phoe calls as a predctor varable Data were collected over a perod of 35 days ad are stored the fle tradesxls a Use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of b 0 ad b ths problem c Predct the umber of trades executed for a day whch the umber of comg calls s,000 d Should you use the model to predct the umber of trades executed for a day whch the umber of comg calls s 5,000? Why or why ot? e Determe the coeffcet of determato, r, ad expla ts meag ths problem f Plot the resduals agast the umber of comg calls ad also agast the days Is there ay evdece of a patter the resduals wth ether of these varables? Expla g Determe the Durb-Watso statstc for these data h Based o the results of (f ) ad (g), s there reaso to questo the valdty of the model? Expla At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the volume of trade executos ad the umber of comg calls? j Costruct a 95% cofdece terval estmate of the mea umber of trades executed for days whch the umber of comg calls s,000 k Costruct a 95% predcto terval of the umber of trades executed for a partcular day whch the umber of comg calls s,000 l Costruct a 95% cofdece terval estmate of the populato slope mbased o the results of (a) through (l), do you thk the brokerage house should focus o a strategy of creasg the total umber of comg calls or o a strategy that reles o tradg by a small umber of heavy traders? Expla 376 You wat to develop a model to predct the sellg prce of homes based o assessed value A sample of 30

48 558 CHAPTER THIRTEEN Smple Lear Regresso recetly sold sgle-famly houses a small cty s selected to study the relatoshp betwee sellg prce ( thousads of dollars) ad assessed value ( thousads of dollars) The houses the cty had bee reassessed at full value oe year pror to the study The results are the fle housexls (Ht: Frst, determe whch are the depedet ad depedet varables) a Costruct a scatter plot ad, assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the sellg prce for a house whose assessed value s $70,000 d Determe the coeffcet of determato, r, ad terpret ts meag ths problem e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee sellg prce ad assessed value? g Costruct a 95% cofdece terval estmate of the mea sellg prce for houses wth a assessed value of $70,000 h Costruct a 95% predcto terval of the sellg prce of a dvdual house wth a assessed value of $70,000 Costruct a 95% cofdece terval estmate of the populato slope 377 You wat to develop a model to predct the assessed value of houses, based o heatg area A sample of 5 sgle-famly houses s selected a cty The assessed value ( thousads of dollars) ad the heatg area of the houses ( thousads of square feet) are recorded, wth the followg results, stored the fle housexls: Assessed Heatg Area of Dwellg House Value ($000) (Thousads of Square Feet) (Ht: Frst, determe whch are the depedet ad depedet varables) a Costruct a scatter plot ad, assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the assessed value for a house whose heatg area s,750 square feet d Determe the coeffcet of determato, r, ad terpret ts meag ths problem e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee assessed value ad heatg area? g Costruct a 95% cofdece terval estmate of the mea assessed value for houses wth a heatg area of,750 square feet h Costruct a 95% predcto terval of the assessed value of a dvdual house wth a heatg area of,750 square feet Costruct a 95% cofdece terval estmate of the populato slope 378 The drector of graduate studes at a large college of busess would lke to predct the grade pot average (GPA) of studets a MBA program based o the Graduate Maagemet Admsso Test (GMAT) score A sample of 0 studets who had completed years the program s selected The results are stored the fle gpgmatxls: GMAT GMAT Observato Score GPA Observato Score GPA (Ht: Frst, determe whch are the depedet ad depedet varables) a Costruct a scatter plot ad, assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the GPA for a studet wth a GMAT score of 600 d Determe the coeffcet of determato, r, ad terpret ts meag ths problem e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model

49 Chapter Revew Problems 559 f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee GMAT score ad GPA? g Costruct a 95% cofdece terval estmate of the mea GPA of studets wth a GMAT score of 600 h Costruct a 95% predcto terval of the GPA for a partcular studet wth a GMAT score of 600 Costruct a 95% cofdece terval estmate of the populato slope 379 The maager of the purchasg departmet of a large bakg orgazato would lke to develop a model to predct the amout of tme t takes to process voces Data are collected from a sample of 30 days, ad the umber of voces processed ad completo tme, hours, s stored the fle vocexls (Ht: Frst, determe whch are the depedet ad depedet varables) a Assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the amout of tme t would take to process 50 voces d Determe the coeffcet of determato, r, ad terpret ts meag e Plot the resduals agast the umber of voces processed ad also agast tme f Based o the plots (e), does the model seem approprate? g Compute the Durb-Watso statstc ad, at the 005 level of sgfcace, determe whether there s ay autocorrelato the resduals h Based o the results of (e) through (g), what coclusos ca you reach cocerg the valdty of the model? At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the amout of tme ad the umber of voces processed? j Costruct a 95% cofdece terval estmate of the mea amout of tme t would take to process 50 voces k Costruct a 95% predcto terval of the amout of tme t would take to process 50 voces o a partcular day 380 O Jauary 8, 986, the space shuttle Challeger exploded, ad seve astroauts were klled Pror to the lauch, the predcted atmospherc temperature was for freezg weather at the lauch ste Egeers for Morto Thokol (the maufacturer of the rocket motor) prepared charts to make the case that the lauch should ot take place due to the cold weather These argumets were rejected, ad the lauch tragcally took place Upo vestgato after the tragedy, experts agreed that the dsaster occurred because of leaky rubber O-rgs that dd ot seal properly due to the cold temperature Data dcatg the atmospherc temperature at the tme of 3 prevous lauches ad the O-rg damage dex are stored the fle o-rgxls: Temperature O-Rg Flght Number ( F) Damage Idex B C 63 4-D G A B C 53 5-D F G I J A B C 58 4 Note: Data from flght 4 s omtted due to ukow O-rg codto Source: Extracted from Report of the Presdetal Commsso o the Space Shuttle Challeger Accdet, Washgto, DC, 986, Vol II (H H3) ad Vol IV (664), ad Post Challeger Evaluato of Space Shuttle Rsk Assessmet ad Maagemet, Washgto, DC, 988, pp a Costruct a scatter plot for the seve flghts whch there was O-rg damage (O-rg damage dex 0) What coclusos, f ay, ca you draw about the relatoshp betwee atmospherc temperature ad O-rg damage? b Costruct a scatter plot for all 3 flghts c Expla ay dffereces the terpretato of the relatoshp betwee atmospherc temperature ad O-rg damage (a) ad (b) d Based o the scatter plot (b), provde reasos why a predcto should ot be made for a atmospherc temperature of 3 F, the temperature o the morg of the lauch of the Challeger e Although the assumpto of a lear relatoshp may ot be vald, ft a smple lear regresso model to predct O-rg damage, based o atmospherc temperature f Iclude the predcto le foud (e) o the scatter plot developed (b) g Based o the results of (f ), do you thk a lear model s approprate for these data? Expla h Perform a resdual aalyss What coclusos do you reach?

50 560 CHAPTER THIRTEEN Smple Lear Regresso 38 Crazy Dave, a well-kow baseball aalyst, would lke to study varous team statstcs for the 005 baseball seaso to determe whch varables mght be useful predctg the umber of ws acheved by teams durg the seaso He has decded to beg by usg a team s eared ru average (ERA), a measure of ptchg performace, to predct the umber of ws The data for the 30 Major League Baseball teams are the fle bb005xls (Ht: Frst, determe whch are the depedet ad depedet varables) a Assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the umber of ws for a team wth a ERA of 450 d Compute the coeffcet of determato, r, ad terpret ts meag e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the umber of ws ad the ERA? g Costruct a 95% cofdece terval estmate of the mea umber of ws expected for teams wth a ERA of 450 h Costruct a 95% predcto terval of the umber of ws for a dvdual team that has a ERA of 450 Costruct a 95% cofdece terval estmate of the slope j The 30 teams costtute a populato I order to use statstcal ferece, as (f) through (), the data must be assumed to represet a radom sample What populato would ths sample be drawg coclusos about? k What other depedet varables mght you cosder for cluso the model? 38 College football players tryg out for the NFL are gve the Woderlc stadardzed tellgece test The data the fle woderlcxls cotas the average Woderlc scores of football players tryg out for the NFL ad the graduato rates for football players at selected schools (extracted from S Walker, The NFL s Smartest Team, The Wall Street Joural, September 30, 005, pp W, W0) You pla to develop a regresso model to predct the Woderlc scores for football players tryg out for the NFL, based o the graduato rate of the school they atteded a Assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the Woderlc score for football players tryg out for the NFL from a school that has a graduato rate of 50% d Compute the coeffcet of determato, r, ad terpret ts meag e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the Woderlc score for a football player tryg out for the NFL from a school ad the school s graduato rate? g Costruct a 95% cofdece terval estmate of the mea Woderlc score for football players tryg out for the NFL from a school that has a graduato rate of 50% h Costruct a 95% predcto terval of the Woderlc score for a football player tryg out for the NFL from a school that has a graduato rate of 50% Costruct a 95% cofdece terval estmate of the slope 383 College basketball s bg busess, wth coaches salares, reveues, ad expeses mllos of dollars The data the fle colleges-basketballxls cotas the coaches salares ad reveues for college basketball at selected schools a recet year (extracted from R Adams, Pay for Playoffs, The Wall Street Joural, March, 006, pp P, P8) You pla to develop a regresso model to predct a coach s salary based o reveue a Assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the coach s salary for a school that has reveue of $7 mllo d Compute the coeffcet of determato, r, ad terpret ts meag e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the coach s salary for a school ad reveue? g Costruct a 95% cofdece terval estmate of the mea salary of coaches at schools that have reveue of $7 mllo h Costruct a 95% predcto terval of the coach s salary for a school that has reveue of $7 mllo Costruct a 95% cofdece terval estmate of the slope 384 Durg the fall harvest seaso the Uted States, pumpks are sold large quattes at farm stads Ofte, stead of weghg the pumpks pror to sale, the farm stad operator wll just place the pumpk the approprate crcular cutout o the couter Whe asked why ths was doe, oe farmer repled, I ca tell the weght of the pumpk from ts crcumferece To determe whether ths was really true, a sample of 3 pumpks were mea-

51 Chapter Revew Problems 56 sured for crcumferece ad weghed, wth the followg results, stored the fle pumpkxls: Crcumferece Weght Crcumferece Weght (cm) (Grams) (cm) (Grams) 50,00 57,000 55,000 66,500 54, ,600 5, , ,00 5, ,500 5,500 47,400 50,500 5,500 49,600 63,500 60, ,00 43,000 a Assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the slope, b, ths problem c Predct the mea weght for a pumpk that s 60 cetmeters crcumferece d Do you thk t s a good dea for the farmer to sell pumpks by crcumferece stead of weght? Expla e Determe the coeffcet of determato, r, ad terpret ts meag f Perform a resdual aalyss for these data ad determe the adequacy of the ft of the model g At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the crcumferece ad the weght of a pumpk? h Costruct a 95% cofdece terval estmate of the populato slope, β Costruct a 95% cofdece terval estmate of the populato mea weght for pumpks that have a crcumferece of 60 cetmeters j Costruct a 95% predcto terval of the weght for a dvdual pumpk that has a crcumferece of 60 cetmeters 385 Ca demographc formato be helpful predctg sales of sportg goods stores? The data stored the fle sportgxls are the mothly sales totals from a radom sample of 38 stores a large cha of atowde sportg goods stores All stores the frachse, ad thus wth the sample, are approxmately the same sze ad carry the same merchadse The couty or, some cases, coutes whch the store draws the majorty of ts customers s referred to here as the customer base For each of the 38 stores, demographc formato about the customer base s provded The data are real, but the ame of the frachse s ot used, at the request of the compay The varables the data set are Sales Latest oe-moth sales total (dollars) Age Meda age of customer base (years) HS Percetage of customer base wth a hgh school dploma College Percetage of customer base wth a college dploma Growth Aual populato growth rate of customer base over the past 0 years Icome Meda famly come of customer base (dollars) a Costruct a scatter plot, usg sales as the depedet varable ad meda famly come as the depedet varable Dscuss the scatter dagram b Assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b c Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem d Compute the coeffcet of determato, r, ad terpret ts meag e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the depedet varable ad the depedet varable? g Costruct a 95% cofdece terval estmate of the slope ad terpret ts meag 386 For the data of Problem 385, repeat (a) through (g), usg meda age as the depedet varable 387 For the data of Problem 385, repeat (a) through (g), usg hgh school graduato rate as the depedet varable 388 For the data of Problem 385, repeat (a) through (g), usg college graduato rate as the depedet varable 389 For the data of Problem 385, repeat (a) through (g), usg populato growth as the depedet varable 390 Zagat s publshes restaurat ratgs for varous locatos the Uted States The data fle restauratsxls cotas the Zagat ratg for food, decor, servce, ad the prce per perso for a sample of 50 restaurats located a urba area (New York Cty) ad 50 restaurats located a suburb of New York Cty Develop a regresso model to predct the prce per perso, based o a varable that represets the sum of the ratgs for food, decor, ad servce Source: Extracted from Zagat Survey 00 New York Cty Restaurats ad Zagat Survey 00 00, Log Islad Restaurats a Assumg a lear relatoshp, use the least-squares method to compute the regresso coeffcets b 0 ad b b Iterpret the meag of the Y tercept, b 0, ad the slope, b, ths problem c Use the predcto le developed (a) to predct the prce per perso for a restaurat wth a summated ratg of 50 d Compute the coeffcet of determato, r, ad terpret ts meag

52 56 CHAPTER THIRTEEN Smple Lear Regresso e Perform a resdual aalyss o your results ad determe the adequacy of the ft of the model f At the 005 level of sgfcace, s there evdece of a lear relatoshp betwee the prce per perso ad the summated ratg? g Costruct a 95% cofdece terval estmate of the mea prce per perso for all restaurats wth a summated ratg of 50 h Costruct a 95% predcto terval of the prce per perso for a restaurat wth a summated ratg of 50 Costruct a 95% cofdece terval estmate of the slope j How useful do you thk the summated ratg s as a predctor of prce? Expla 39 Refer to the dscusso of beta values ad market models Problem 349 o pages Oe hudred weeks of data, edg the week of May, 006, for the S&P 500 ad three dvdual stocks are cluded the data fle sp500xls Note that the weekly percetage chage for both the S&P 500 ad the dvdual stocks s measured as the percetage chage from the prevous week s closg value to the curret week s closg value The varables cluded are Week Curret week SP500 Weekly percetage chage the S&P 500 Idex WALMART Weekly percetage chage stock prce of Wal-Mart Stores, Ic TARGET Weekly percetage chage stock prce of the Target Corporato SARALEE Weekly percetage chage stock prce of the Sara Lee Corporato Source: Extracted from faceyahoocom, May 3, 006 a Estmate the market model for Wal-Mart Stores Ic (Ht: Use the percetage chage the S&P 500 Idex as the depedet varable ad the percetage chage Wal- Mart Stores, Ic s stock prce as the depedet varable) b Iterpret the beta value for Wal-Mart Stores, Ic c Repeat (a) ad (b) for Target Corporato d Repeat (a) ad (b) for Sara Lee Corporato e Wrte a bref summary of your fdgs 39 The data fle retursxls cotas the stock prces of four compaes, collected weekly for 53 cosecutve weeks, edg May, 006 The varables are Week Closg date for stock prces MSFT Stock prce of Mcrosoft, Ic Ford Stock prce of Ford Motor Compay GM Stock prce of Geeral Motors, Ic IAL Stock prce of Iteratoal Alumum, Ic Source: Extracted from faceyahoocom, May 3, 006 a Calculate the correlato coeffcet, r, for each par of stocks (There are sx of them) b Iterpret the meag of r for each par c Is t a good dea to have all the stocks a dvdual s portfolo be strogly postvely correlated amog each other? Expla 393 Is the daly performace of stocks ad bods correlated? The data fle stocks&bodsxls cotas formato cocerg the closg value of the Dow Joes Idustral Average ad the Vaguard Log-Term Bod Idex Fud for 60 cosecutve busess days, edg May 30, 006 The varables cluded are Date Curret day Bods Closg prce of Vaguard Log-Term Bod Idex Fud Stocks Closg prce of the Dow Joes Idustral Average Source: Extracted from faceyahoocom, May 3, 006 a Compute ad terpret the correlato coeffcet, r, for the varables Stocks ad Bods b At the 005 level of sgfcace, s there a relatoshp betwee these two varables? Expla Report Wrtg Exercses 394 I Problems o page 56, you developed regresso models to predct mothly sales at a sportg goods store Now, wrte a report based o the models you developed Apped to your report all approprate charts ad statstcal formato Maagg the Sprgvlle Herald To esure that as may tral subscrptos as possble are coverted to regular subscrptos, the Herald marketg departmet works closely wth the dstrbuto departmet to accomplsh a smooth tal delvery process for the tral subscrpto customers To assst ths effort, the marketg departmet eeds to accurately forecast the umber of ew regular subscrptos for the comg moths A team cosstg of maagers from the marketg ad dstrbuto departmets was coveed to develop a better method of forecastg ew subscrptos Prevously, after examg ew subscrpto data for the pror three moths, a group of three maagers would develop a subjectve forecast of the umber of ew subscrptos Laure Hall, who was recetly hred by the compay to provde specal sklls quattatve forecastg methods, suggested that the departmet look for factors that mght help predctg ew subscrptos Members of the team foud that the forecasts the past year had bee partcularly accurate because some moths, much more tme was spet o telemarketg tha

53 Refereces 563 other moths I partcular, the past moth, oly,055 hours were completed because callers were busy durg the frst week of the moth attedg trag sessos o the persoal but formal greetg style ad a ew stadard presetato gude (see Maagg the Sprgvlle Herald Chapter ) Laure collected data (stored the fle sh3xls) for the umber of ew subscrptos ad hours spet o telemarketg for each moth for the past two years EXERCISES SH3 What crtcsm ca you make cocerg the method of forecastg that volved takg the ew subscrptos data for the pror three moths as the bass for future projectos? SH3 What factors other tha umber of telemarketg hours spet mght be useful predctg the umber of ew subscrptos? Expla SH33 a Aalyze the data ad develop a regresso model to predct the mea umber of ew subscrptos for a moth, based o the umber of hours spet o telemarketg for ew subscrptos b If you expect to sped,00 hours o telemarketg per moth, estmate the mea umber of ew subscrptos for the moth Idcate the assumptos o whch ths predcto s based Do you thk these assumptos are vald? Expla c What would be the dager of predctg the umber of ew subscrptos for a moth whch,000 hours were spet o telemarketg? Web Case Apply your kowledge of smple lear regresso ths Web Case, whch exteds the Suflowers Apparel Usg Statstcs scearo from ths chapter Leasg agets from the Tragle Mall Maagemet Corporato have suggested that Suflowers cosder several locatos some of Tragle s ewly reovated lfestyle malls that cater to shoppers wth hgher-tha-mea dsposable come Although the locatos are smaller tha the typcal Suflowers locato, the leasg agets argue that hgher-tha-mea dsposable come the surroudg commuty s a better predctor of hgher sales tha store sze The leasg agets mata that sample data from 4 Suflowers stores prove that ths s true Revew the leasg agets proposal ad supportg documets that descrbe the data at the compay s Web ste, wwwprehallcom/sprgvlle/tragle_suflowerhtm, (or ope ths Web case fle from the Studet CD-ROM s Web Case folder), ad the aswer the followg: Should mea dsposable come be used to predct sales based o the sample of 4 Suflowers stores? Should the maagemet of Suflowers accept the clams of Tragle s leasg agets? Why or why ot? 3 Is t possble that the mea dsposable come of the surroudg area s ot a mportat factor leasg ew locatos? Expla 4 Are there ay other factors ot metoed by the leasg agets that mght be relevat to the store leasg decso? REFERENCES Ascombe, F J, Graphs Statstcal Aalyss, The Amerca Statstca 7 (973): 7 Hoagl, D C, ad R Welsch, The Hat Matrx Regresso ad ANOVA, The Amerca Statstca 3 (978): 7 3 Hockg, R R, Developmets Lear Regresso Methodology: , Techometrcs 5 (983): Kuter, M H, C J Nachtshem, J Neter, ad W L, Appled Lear Statstcal Models, 5th ed (New York: McGraw-Hll/Irw, 005) 5 Mcrosoft Excel 007 (Redmod, WA: Mcrosoft Corp, 007)

54 LEVIMC3_034057QXD 564 //07 4:4 PM Page 564 EXCEL COMPANION to Chapter 3 Excel Compao to Chapter 3 E3 PERFORMING SIMPLE LINEAR REGRESSION ANALYSES You perform a smple lear regresso aalyss by ether usg the PHStat Smple Lear Regresso procedure or by usg the ToolPak Regresso procedure PHStat performs the regresso aalyss, usg the ToolPak Regresso procedure Therefore, the worksheet produced does ot dyamcally chage f you chage your data (Reru the procedure to create revsed results) The three Output Optos avalable the PHStat dalog box ehace the ToolPak procedure ad are explaed Sectos E3, E34, ad E35 Usg PHStat Smple Lear Regresso Usg ToolPak Regresso Ope to the worksheet that cotas the data for the regresso aalyss Select PHStat Regresso Smple Lear Regresso I the procedure s dalog box (show below), eter the cell rage of the Y varable as the Y Varable Cell Rage ad the cell rage of the X varable as the X Varable Cell Rage Clck Frst cells both rages cota label ad eter a value for the Cofdece level for regresso coeffcets Clck the Regresso Statstcs Table ad the ANOVA ad Coeffcets Table Regresso Tool Output Optos, eter a ttle as the Ttle, ad clck OK Ope to the worksheet that cotas the data for the regresso aalyss Select Tools Data Aalyss, select Regresso from the Data Aalyss lst, ad clck OK I the procedure s dalog box (show below), eter the cell rage of the Y varable data as the Iput Y Rage ad eter the cell rage of the X varable data as the Iput X Rage Clck Labels, clck Cofdece Level ad eter a value ts box, ad the clck OK Results appear o a ew worksheet E3 CREATING SCATTER PLOTS AND ADDING A PREDICTION LINE You use Excel chartg features to create a scatter plot ad add a predcto le to that plot If you select the Scatter Dagram output opto of the PHStat Smple Lear Regresso procedure (see Secto E3), you ca skp to the Addg a Predcto Le secto that apples to the Excel verso you use

55 LEVIMC3_034057QXD //07 4:4 PM Page 565 E3: Creatg Scatter Plots ad Addg a Predcto Le 565 Creatg a Scatter Plot Use ether the Secto E structos to create a scatter plot (see page 93) or use the Secto E3 structos Usg PHStat Smple Lear Regresso, but clckg Scatter Dagram before you clck OK Addg a Predcto Le (97 003) Ope to the chart sheet that cotas your scatter plot ad select Chart Add Tredle I the Add Tredle dalog box (see Fgure E3), clck the Type tab ad the clck Lear Clck the Optos tab ad select the Automatc opto Clck Dsplay equato o chart ad Dsplay R-squared value o chart ad the clck OK If you have cluded a label as part of your data rage, you wll see that label dsplayed place of Seres ths dalog box FIGURE E3 Format Tredle dalog box (007) relocate the X axs to the bottom of the chart, ope to the chart, rght-clck the Y axs ad select Format Axs from the shortcut meu If you use Excel , select the Scale tab the Format Axs dalog box (see Fgure E33), ad eter the value foud the Mmum box (-6 Fgure E33) as the Value (X) axs Crosses at value ad clck OK (As you eter ths value, the check box for ths etry s cleared automatcally) FIGURE E3 Add Tredle dalog box (97 003) Addg a Predcto Le (007) Ope to the chart sheet that cotas your scatter plot ad select Layout Tredle ad the Tredle gallery, select More Tredle Optos I the Tredle Optos pael of the Format Tredle dalog box (see Fgure E3), select the Lear opto, clck Dsplay equato o chart ad Dsplay R-squared value o chart, ad clck Close Relocatg a X Axs If there are Y values o a resdual plot or scatter plot that are less tha zero, Mcrosoft Excel places the X axs at the pot Y = 0, possbly obscurg some of the data pots To FIGURE E33 Format Axs dalog box (97 003)

56 566 EXCEL COMPANION to Chapter 3 If you use Excel 007, the Axs Optos pael of the Format Axs dalog box (see Fgure E34), select the Axs value opto, chage ts default value of 00 (show Fgure E34) to a value less tha the mmum Y value, ad clck Close FIGURE E34 Format Axs dalog box (007) E33 PERFORMING RESIDUAL ANALYSES You modfy the procedures of Secto E3 to perform a resdual aalyss If you use the PHStat Smple Lear Regresso procedure, clck all the Regresso Tool output optos (Regresso Statstcs Table, ANOVA ad Coeffcets Table, Resduals Table, ad Resdual Plot) If you use the ToolPak Regresso procedure, clck Resduals ad Resdual Plots before clckg OK If you eed to relocate a X axs to the bottom of a resdual plot, revew the Relocatg a X Axs part of Secto E3 E34 COMPUTING THE DURBIN- WATSON STATISTIC You compute the Durb-Watso Statstc by ether usg the PHStat Smple Lear Regresso procedure or by usg a several-step process that uses the Durb-Watsoxls workbook Usg PHStat Smple Lear Regresso Use the Secto E3 structos Usg PHStat Smple Lear Regresso, but clckg Durb-Watso Statstc before you clck OK Choosg the Durb- Watso Statstc causes PHStat to create a resduals table, eve f you dd ot check the Resduals Table Regresso Tool output opto The Durb-Watso Statstc output opto creates a ew Durb-Watso worksheet smlar to the oe show Fgure 36 o page 536 Ths worksheet refereces cells the regresso results worksheet that s also created by the procedure If you delete the regresso results worksheet, the DurbWatso worksheet dsplays a error message Usg Durb-Watsoxls Ope to the DurbWatso worksheet of the Durb-Watsoxls workbook Ths worksheet (see Fgure 36 o page 536) uses the SUMXMY (cell rage, cell rage ) fucto cell B3 to compute the sum of squared dfferece of the resduals, ad the SUMSQ (resduals cell rage) fucto cell B4 to compute the sum of squared resduals for the Secto 36 package delvery store example By settg cell rage to the cell rage of the frst resdual through the secod-to-last resdual ad cell rage to the cell rage of the secod resdual through the last resdual, you ca get SUMXMY to compute the squared dfferece betwee two successve resduals, whch s the umerator term of Equato (35) Because resduals appear a regresso results worksheet, cell refereces used the SUMXMY fucto must refer to the regresso results worksheet by ame I the Durb-Watso workbook, the SLR worksheet cotas the smple lear regresso aalyss for the Secto 36 package delvery example The resduals appear the cell rage C5:C39 Therefore, cell rage s set to SLR!C5:C38, ad cell rage s set to SLR!C6:C39 Ths makes the cell B3 formula =SUMXMY(SLR!C6:C39, SLR!C5:C38) The cell B4 formula, whch also must refer to the SLR worksheet, s =SUMSQ(SLR!C5:C39) To adapt the Durb-Watso workbook to other problems, frst create a smple lear regresso results worksheet that cotas resdual output ad copy that worksheet to the Durb-Watso workbook The ope to the Durb-Watso worksheet ad edt the formulas cells B3 ad B4 so that they refer to the correct cell rages o your regresso worksheet Fally, delete the o-logereeded SLR worksheet E35 ESTIMATING THE MEAN OF Y AND PREDICTING Y VALUES You compute a cofdece terval estmate for the mea respose ad the predcto terval for a dvdual respose ether by selectg the PHStat Smple Lear Regresso procedure or by makg etres the CIEadPIforSLRxls workbook

57 E36: Example: Suflowers Apparel Data 567 FIGURE E35 DataCopy worksheet (frst sx rows) Usg PHStat Smple Lear Regresso Use the Secto E3 structos Usg PHStat Smple Lear Regresso, but before you clck OK, clck Cofdece ad Predcto Iterval for X = ad eter a X value ts box (see below) The eter a value for the Cofdece level for terval estmates ad clck OK Cells B8, B, B, ad B5 cota formulas that referece dvdual cells o a DataCopy worksheet Ths worksheet, the frst sx rows of whch are show Fgure E35, cotas a copy of the regresso data colums A ad B ad a formula colum C that squares the dfferece betwee each X ad X The worksheet also computes the sample sze, the sample mea, the sum of the squared dffereces [SSX Equato (30) o page 546], ad the predcted Y value cells F, F3, F4, ad F5 The cell F5 formula uses the fucto TREND (Y varable cell rage, Xvarable cell rage, X value) to calculate the predcted Y value Because the formula uses the X value that has bee etered o the CIEadPI worksheet, the X value the cell F5 formula s set to CIEadPI!B4 Because the DataCopy ad CIEadPI worksheets referece each other, you should cosder these worksheets a matched par that should ot be broke up To adapt these worksheets to other problems, frst create a smple lear regresso results worksheet The, trasfer the stadard error value, always foud the regresso results worksheet cell B7, to cell B3 of the CIEadPI worksheet Chage, as s ecessary, the X Value ad the cofdece level cells B4 ad B5 of the CIEadPI worksheet Next, ope to the DataCopy worksheet, ad f your sample sze s ot 4, follow the structos foud the worksheet Eter the problem s X values colum A ad Y values colum B Fally, retur to the CIEadPI worksheet to exame ts updated results PHStat places the cofdece terval estmate ad predcto terval o a ew worksheet smlar to the oe show Fgure 3 o page 549 (PHStat also creates a DataCopy worksheet that s dscussed the ext part of ths secto) Usg CIEadPIforSLRxls Ope to the CIEadPI worksheet of the CIEadPIforSLRxls workbook Ths worksheet (show Fgure 3 o page 549) uses the fucto TINV(- cofdece level, degrees of freedom) to determe the t value ad compute the cofdece terval estmate ad predcto terval for the Secto 38 Suflower s Apparel example E36 EXAMPLE: SUNFLOWERS APPAREL DATA Ths secto shows you how to use PHStat or Basc Excel to perform a regresso aalyss for Suflowers Apparel usg the square footage ad aual sales data stored the stexls workbook Usg PHStat Ope to the Data worksheet of the stexls workbook Select PHStat Regresso Smple Lear Regresso I the procedure s dalog box (see Fgure E36), eter C:C5 as the Y Varable Cell Rage ad B:B5 as the X Varable Cell Rage Clck Frst cells both rages

58 LEVIMC3_034057QXD 568 //07 4:43 PM Page 568 EXCEL COMPANION to Chapter 3 FIGURE E37 Completed Normal Probablty Plot dalog box You coclude that all assumptos are vald ad that you ca use ths smple lear regresso model for the Suflowers Apparel data You ca ow ope to the SLR worksheet to vew the detals of the aalyss or ope to the Estmate worksheet to make fereces about the mea of Y ad the predcto of dvdual values of Y FIGURE E36 Completed Smple Lear Regresso dalog box cota label ad eter a value for the Cofdece level for regresso coeffcets Clck the Regresso Statstcs Table, ANOVA ad Coeffcets Table, Resduals Table, ad Resdual Plot Regresso Tool Output Optos Eter Ste Selecto Aalyss as the Ttle ad clck Scatter Dagram Clck Cofdece ad Predcto Iterval for X= ad eter 4 ts box Eter 95 the Cofdece level for terval estmates box Clck OK to execute the procedure To evaluate the assumpto of learty, you revew the Resdual Plot for X chart sheet Note that there s o apparet patter or relatoshp betwee the resduals ad X varable To evaluate the ormalty assumpto, create a ormal probablty plot Wth your workbook ope to the SLR worksheet, select PHStat Probablty & Prob Dstrbutos Normal Probablty Plot I the procedure s dalog box (see Fgure E37), eter C4:C38 as the Varable Cell Rage ad clck Frst cell cotas label Eter Normal Probablty Plot as the Ttle ad clck OK I the NormalPlot chart sheet, observe that the data do ot appear to depart substatally from a ormal dstrbuto To evaluate the assumpto of equal varaces, revew the Resdual Plot for X chart sheet Note that there do ot appear to be major dffereces the varablty of the resduals Usg Basc Excel Ope to the Data worksheet of the stexls workbook Select Tools Data Aalyss (97 003) or Data Data Aalyss (007) Select Regresso from the Data Aalyss lst, ad clck OK I the procedure s dalog box (see Fgure E38), eter C:C5 as the Iput Y Rage ad eter B:B5 as the Iput X Rage Clck Labels, clck Cofdece Level ad eter 95 ts box, ad clck Resduals Clck OK to execute the procedure FIGURE E38 Completed Regresso dalog box

59 E36: Example: Suflowers Apparel Data 569 To evaluate the assumpto of learty, you plot the resduals agast the square feet (depedet) varable To smplfy creatg ths plot, ope to the Data worksheet ad copy the square feet cell rage B:B5 to cell E The copy the cell rage of the resduals, C4:C38 o the SLR worksheet, to cell F of the Data worksheet Wth your workbook ope to the Data worksheet, use the Secto E3 structos o pages to create a scatter plot (Use E:F5 as the Data rage (Excel ) or as the cell rage of the X ad Y varables (Excel 007) whe creatg the scatter plot) Revew the scatter plot Observe that there s o apparet patter or relatoshp betwee the resduals ad X varable You coclude that the learty assumpto holds You ow evaluate the ormalty assumpto by creatg a ormal probablty plot Create a Plot worksheet, usg the model worksheet the NPPxls workbook as your gude I a ew worksheet, eter Rak cell A ad the eter the seres through 4 cells A:A5 Eter Proporto cell B ad eter the formula =A/5 cell B Next, eter Z Value cell C ad the formula =NORMSINV(B) cell C Copy the resduals (cludg ther colum headg) to the cell rage D:D5 Select the formulas cell rage B:C ad copy them dow through row 5 Ope to the probablty plot ad observe that the data do ot appear to depart substatally from a ormal dstrbuto To evaluate the assumpto of equal varace, retur to the scatter plot of the resduals ad the X varable that you already developed Observe that there do ot appear to be major dffereces the varablty of the resduals You coclude that all assumptos are vald ad that you ca use ths smple lear regresso model for the Suflowers Apparel data You ca ow evaluate the detals of the regresso results worksheet If you are terested makg fereces about the mea of Y ad the predcto of dvdual values of Y, ope the CIEadPIforSLRxls workbook (Usually, you would have to frst make adjustmets to the DataCopy worksheet, as dscussed Secto E35, but ths workbook already cotas the etres for the Suflowers Apparel aalyss) Ope to the CIEadPI worksheet to make fereces about the mea of Y ad the predcto of dvdual values of Y

60