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?