Business Statistics: A Decision-Making Approach, 6e

Transcription

1 Chaper 15 Suden Lecure Noes 15-1 Business Saisics: A Decision-Making Approach 6 h Ediion Chaper 16 Analzing and Forecasing Time-Series Daa Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-1 Chaper Goals Afer compleing his chaper, ou should be able o: Develop and implemen basic forecasing models Idenif he componens presen in a ime series Compue and inerpre basic index numbers Use smoohing-based forecasing models, including single and double exponenial smoohing Appl rend-based forecasing models, including linear rend, nonlinear rend, and seasonall adjused rend Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-2 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

2 Chaper 15 Suden Lecure Noes 15-2 The Imporance of Forecasing Governmens forecas unemplomen, ineres raes, and expeced revenues from income axes for polic purposes Markeing execuives forecas demand, sales, and consumer preferences for sraegic planning College adminisraors forecas enrollmens o plan for faciliies and for facul recruimen Reail sores forecas demand o conrol invenor levels, hire emploees and provide raining Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-3 Wha is he difference beween planning and forecasing? Planning is he process of deermining how o deal wih he fuure. Forecasing is he process of predicing he iming and magniude of fuure evens, predicing wha he fuure will be like. Manufacuring firms mus plan heir producion: an exercise known as aggregae producion planning in which he firm saes how man unis o produce on a period b period basis and wha level of emplomen o have over ha ime period. A forecas of he firm s demand is a necessar inpu o he producion planning process. Likewise, service organizaions will use a forecas in heir budgeing and planning aciviies. Shor erm demand forecass ma be used as one inpu o deermine he number of workers o schedule for a paricular shif. Elecric uili companies will use a demand forecas o plan heir long-erm capaci requiremens, as well as plan and prepare for shor-erm needs. Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-4 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

3 Inflaion Rae (%) Chaper 15 Suden Lecure Noes 15-3 Time-Series Daa Numerical daa obained a regular ime inervals The ime inervals can be annuall, quarerl, dail, hourl, ec. Example: Year: : Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-5 Time Series Plo A ime-series plo is a wo-dimensional plo of ime series daa he verical axis measures he variable of ineres he horizonal axis corresponds o he ime periods U.S. Inflaion Rae Year Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-6 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

4 Chaper 15 Suden Lecure Noes 15-4 Time-Series Componens Time-Series Trend Componen Seasonal Componen Cclical Componen Random Componen Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-7 Trend Componen Long-run increase or decrease over ime (overall upward or downward movemen) Daa aken over a long period of ime Time Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-8 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

5 Chaper 15 Suden Lecure Noes 15-5 Trend Componen Trend can be upward or downward Trend can be linear or non-linear (coninued) Downward linear rend Time Time Upward nonlinear rend Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15-9 Seasonal Componen Shor-erm regular wave-like paerns Observed wihin 1 ear Ofen monhl or quarerl Winer Summer Spring Fall Time (Quarerl) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

6 Chaper 15 Suden Lecure Noes 15-6 A seasonal componen is a paern in he ime series ha repeas iself wih he same period of recurrence. While we ofen hink of seasonal effecs as being associaed wih he seasons (spring, summer, fall, winer) of he ear, he seasonal paern ma be hourl, dail, weekl, or monhl. In fac a seasonal paern can be an repeaing paern where he period of recurrence is a mos one ear. An example of a seasonal componen ha is no associaed wih he seasons is he sales of ickes o a movie heaer. Ticke \sales ma well be higher on Frida and Saurda evenings, han he are on Tuesda and Wednesda afernoons. If his paern repeas iself over ime, he series is said o exhibi a dail seasonal effec. Likewise, phone calls coming o a swichboard ma be higher a cerain hours of he da (beween 9:00 a.m. and 10:00 a.m.) han a oher imes (beween 3:00 p.m. and 4:00 p.m.). If his paern repeas iself in a predicable wa hen we have an hourl seasonal componen. Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Cclical Componen Long-erm wave-like paerns Regularl occur bu ma var in lengh Ofen measured peak o peak or rough o rough 1 Ccle Year Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

7 Chaper 15 Suden Lecure Noes 15-7 Random Componen Unpredicable, random, residual flucuaions Due o random variaions of Naure Accidens or unusual evens Noise in he ime series Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Index Numbers Index numbers allow relaive comparisons over ime Index numbers are repored relaive o a Base Period Index Base period index = 100 b definiion Used for an individual iem or measuremen Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

8 Chaper 15 Suden Lecure Noes 15-8 Index Numbers Simple Index number formula: (coninued) I where I = index number a ime period = value of he ime series a ime 0 = value of he ime series in he base period Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Index Numbers: Example Compan orders from 1995 o 2003: Year Number of Orders Index (base ear = 2000) I (100) Base Year: I 100 (100) I (100) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

9 Chaper 15 Suden Lecure Noes 15-9 Index Numbers: Inerpreaion I (100) I (100) I (100) Orders in 1996 were 90% of base ear orders Orders in 2000 were 100% of base ear orders (b definiion, since 2000 is he base ear) Orders in 2003 were 120% of base ear orders Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Aggregae Price Indexes An aggregae index is used o measure he rae of change from a base period for a group of iems Aggregae Price Indexes Unweighed aggregae price index Weighed aggregae price indexes Paasche Index Lasperes Index Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

10 Chaper 15 Suden Lecure Noes Unweighed Aggregae Price Index Unweighed aggregae price index formula: where I p p 0 (100) I = unweighed aggregae price index a ime p = sum of he prices for he group of iems a ime p 0 = sum of he prices for he group of iems in he base period Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Unweighed Aggregae Price Index Example Auomobile Expenses: Monhl Amouns ($): Year Lease pamen Fuel Repair Toal Index (2001=100) Combined expenses in 2004 were 18.8% higher in 2004 han in 2001 Chan ge % % % % I p 410 (100) (100) p Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

11 Chaper 15 Suden Lecure Noes Weighed Aggregae Price Indexes I Paasche index q p q p 0 (100) I Lasperes index q q 0 0 p p 0 (100) q = weighing percenage a ime q 0 = weighing percenage a base period p = price in ime period p 0 = price in he base period Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Commonl Used Index Numbers Consumer Price Index (CPI): weighed aggregae index similar o Lasperes Index, is based on iems grouped ino caegories (such as food, housing, clohing, ransporaion, medical care, enerainmen, and miscellaneous iem) Producer Price Index (PPI): like he CPI, he PPI is a Lasperes weighed aggregae Index. Sock Marke Indexes Dow Jones Indusrial Average S&P 500 Index NASDAQ Index Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

12 Chaper 15 Suden Lecure Noes Deflaing a Time Series Observed values can be adjused o base ear equivalen Allows uniform comparison over ime Deflaion formula: where adj I (100) adj = adjused ime series value a ime = value of he ime series a ime I = index (such as CPI) a ime Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Deflaing a Time Series: Example Which movie made more mone (in real erms)? Year 1939 Movie Tile Gone Wih he Wind Toal Gross $ Sar Wars Tianic 601 (Toal Gross $ = Toal domesic gross icke receips in $millions) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

13 Chaper 15 Suden Lecure Noes Deflaing a Time Series: Example (coninued) Year 1939 Movie Tile Gone Wih he Wind Toal Gross CPI (base ear = 1984) Gross adjused o 1984 dollars Sar Wars Tianic GWTW adj (100) GWTW made abou wice as much as Sar Wars, and abou 4 imes as much as Tianic when measured in equivalen dollars Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Trend-Based Forecasing Esimae a rend line using regression analsis Year Time Period () () Use ime () as he independen variable: ŷ b0 b1 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

14 sales sales Chaper 15 Suden Lecure Noes Trend-Based Forecasing (coninued) The linear rend model is: Year Time Period () () ŷ rend Year Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Trend-Based Forecasing Forecas for ime period 7: (coninued) Year Time Period () () ?? ŷ (7) Year Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

15 Chaper 15 Suden Lecure Noes Comparing Forecas Values o Acual Daa The forecas error or residual is he difference beween he acual value in ime and he forecas value in ime : Error in ime : e F Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Two common Measures of Fi Measures of fi are used o gauge how well he forecass mach he acual values MSE (mean squared error) Average squared difference beween and F MAD (mean absolue deviaion) Average absolue value of difference beween and F Less sensiive o exreme values Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

16 Residuals Chaper 15 Suden Lecure Noes MSE vs. MAD Mean Square Error MSE ( n F ) 2 Mean Absolue Deviaion MAD n F where: = Acual value a ime F = Prediced value a ime n = Number of ime periods Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Auocorrelaion (coninued) Auocorrelaion is correlaion of he error erms (residuals) over ime Time () Residual Plo Here, residuals show a cclic paern, no random Time () Violaes he regression assumpion ha residuals are random and independen Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

17 Chaper 15 Suden Lecure Noes Tesing for Auocorrelaion The Durbin-Wason Saisic is used o es for auocorrelaion H 0 : ρ = 0 H A : ρ 0 (residuals are no correlaed) (auocorrelaion is presen) Durbin-Wason es saisic: d n 1 (e n 1 e e ) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Tesing for Posiive Auocorrelaion H 0 : ρ = 0 H A : ρ > 0 (posiive auocorrelaion does no exis) (posiive auocorrelaion is presen) Calculae he Durbin-Wason es saisic = d (The Durbin-Wason Saisic can be found using PHSa or Miniab) Find he values d L and d U from he Durbin-Wason able (for sample size n and number of independen variables p) Decision rule: rejec H 0 if d < d L Rejec H 0 Inconclusive Do no rejec H 0 0 d L d U 2 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

18 Chaper 15 Suden Lecure Noes Tesing for Posiive Auocorrelaion Example wih n = 25: (coninued) Excel/PHSa oupu: Durbin-Wason Calculaions Sum of Squared Difference of Residuals Sum of Squared Residuals Durbin-Wason Saisic Time = x R 2 = d n 1 (e e n 1 e 2 ) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Tesing for Posiive Auocorrelaion (coninued) Here, n = 25 and here is one independen variable Using he Durbin-Wason able, d L = 1.29 and d U = 1.45 d = < d L = 1.29, so rejec H 0 and conclude ha significan posiive auocorrelaion exiss Therefore he linear model is no he appropriae model o forecas sales Decision: rejec H 0 since d = < d L Rejec H 0 Inconclusive Do no rejec H 0 d L = d U = Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

19 Chaper 15 Suden Lecure Noes Nonlinear Trend Forecasing A nonlinear regression model can be used when he ime series exhibis a nonlinear rend One form of a nonlinear model: β 0 β 1 2 ε Compare R 2 and s ε o ha of linear model o see if his is an improvemen Can r oher funcional forms o ge bes fi Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Muliplicaive Time-Series Model Used primaril for forecasing Allows consideraion of seasonal variaion Observed value in ime series is he produc of componens T S C I where T = Trend value a ime S = Seasonal value a ime C = Cclical value a ime I = Irregular (random) value a ime Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

20 Chaper 15 Suden Lecure Noes Finding Seasonal Indexes Raio-o-moving average mehod: Begin b removing he seasonal and irregular componens (S and I ), leaving he rend and cclical componens (T and C ) To do his, we need moving averages Moving Average: averages of consecuive ime series values Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Moving Averages Used for smoohing Series of arihmeic means over ime Resul dependen upon choice of L (lengh of period for compuing means) To smooh ou seasonal variaion, L should be equal o he number of seasons For quarerl daa, L = 4 For monhl daa, L = 12 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap 15- Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

21 Chaper 15 Suden Lecure Noes Moving Averages Example: Four-quarer moving average (coninued) Firs average: Q1 Q2 Q3 Q4 Moving average 1 4 Second average: Q2 Q3 Q4 Q5 Moving average 2 4 ec Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Seasonal Daa Quarer ec ec Quarerl Quarer Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

22 Chaper 15 Suden Lecure Noes Calculaing Moving Averages Quarer ec Average Period 4-Quarer Moving Average Each moving average is for a consecuive block of 4 quarers Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Cenered Moving Averages Average periods of 2.5 or 3.5 don mach he original quarers, so we average wo consecuive moving averages o ge cenered moving averages Average Period 4-Quarer Moving Average ec Cenered Period Cenered Moving Average Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

23 Chaper 15 Suden Lecure Noes Calculaing he Raio-o-Moving Average Now esimae he S x I value Divide he acual sales value b he cenered moving average for ha quarer Raio-o-Moving Average formula: S I T C Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Calculaing Seasonal Indexes Quarer Cenered Moving Average ec Raio-o- Moving Average ec Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

24 Chaper 15 Suden Lecure Noes Fall Fall Fall Quarer Calculaing Seasonal Indexes Cenered Moving Average ec Raio-o- Moving Average ec (coninued) Average all of he Fall values o ge Fall s seasonal index Do he same for he oher hree seasons o ge he oher seasonal indexes Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Inerpreing Seasonal Indexes Suppose we ge hese seasonal indexes: Season Seasonal Index Spring Summer Fall Inerpreaion: Spring sales average 82.5% of he annual average sales Summer sales are 31.0% higher han he annual average sales ec Winer = four seasons, so mus sum o 4 Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

25 Chaper 15 Suden Lecure Noes Deseasonalizing The daa is deseasonalized b dividing he observed value b is seasonal index T C I S This smoohs he daa b removing seasonal variaion Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Quarer Deseasonalizing Seasonal Index Deseasonalized ec (coninued) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

26 Chaper 15 Suden Lecure Noes Unseasonalized vs. Seasonalized : Unseasonalized vs. Seasonalized Quarer Deseasonalized Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Forecasing Using Smoohing Mehods Exponenial Smoohing Mehods Single Exponenial Smoohing Double Exponenial Smoohing Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

27 Chaper 15 Suden Lecure Noes Single Exponenial Smoohing A weighed moving average Weighs decline exponeniall Mos recen observaion weighed mos Used for smoohing and shor erm forecasing Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Single Exponenial Smoohing The weighing facor is Subjecivel chosen Range from 0 o 1 Smaller gives more smoohing, larger gives less smoohing The weigh is: Close o 0 for smoohing ou unwaned cclical and irregular componens Close o 1 for forecasing (coninued) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

28 Chaper 15 Suden Lecure Noes Exponenial Smoohing Model Single exponenial smoohing model F 1 F ( F ) or: F 1 (1 ) F where: F +1 = forecas value for period + 1 = acual value for period F = forecas value for period = alpha (smoohing consan) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Exponenial Smoohing Example Suppose we use weigh =.2 Quarer () ec ( ) ec Forecas from prior period NA ec Forecas for nex period (F +1 ) 23 (.2)()+(.8)(23)=26.4 (.2)(25)+(.8)(26.4)=26.12 (.2)(27)+(.8)(26.12)= (.2)(32)+(.8)(26.296)= (.2)(48)+(.8)(27.437)= (.2)(48)+(.8)(31.549)=31.8 (.2)(33)+(.8)(31.8)= (.2)(37)+(.8)(32.872)= (.2)(50)+(.8)(33.697)= ec F 1 = 1 since no prior informaion exiss Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap F 1 (1 )F Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

29 Chaper 15 Suden Lecure Noes vs. Smoohed Seasonal flucuaions have been smoohed NOTE: he smoohed value in his case is generall a lile low, since he rend is upward sloping and he weighing facor is onl Quarer Smoohed Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Double Exponenial Smoohing Double exponenial smoohing is someimes called exponenial smoohing wih rend If rend exiss, single exponenial smoohing ma need adjusmen Add a second smoohing consan o accoun for rend Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

30 Chaper 15 Suden Lecure Noes where: Double Exponenial Smoohing Model C (1 )(C 1 T 1) T (C F 1 C C 1 ) (1 ) T 1 = acual value in ime = consan-process smoohing consan = rend-smoohing consan C = smoohed consan-process value for period T = smoohed rend value for period F +1 = forecas value for period + 1 = curren ime period Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap T Double Exponenial Smoohing Double exponenial smoohing is generall done b compuer Use larger smoohing consans and β when less smoohing is desired Use smaller smoohing consans and β when more smoohing is desired Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.

31 Chaper 15 Suden Lecure Noes Exponenial Smoohing in Excel Use ools / daa analsis / exponenial smoohing The damping facor is (1 - ) Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Chaper Summar Discussed he imporance of forecasing Addressed componen facors presen in he ime-series model Compued and inerpreed index numbers Described leas square rend fiing and forecasing linear and nonlinear models Performed smoohing of daa series moving averages single and double exponenial smoohing Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc. Chap Business Saisics: A Decision-Making Approach, 6e 2005 Prenice-Hall, Inc.