Nonparametric Methods: Goodness-of-Fit Tests

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1 Nonparamtric Mthods: Goodnss-o-Fit Tsts Chaptr 15 McGraw-Hill/Irwin Copyright 013 by Th McGraw-Hill Companis, Inc. All rights rsrvd.

2 LEARNING OBJECTIVES LO 15-1 Conduct a tst o hypothsis comparing an obsrvd st o rquncis to an xpctd distribution. LO 15- List and xplain th charactristics o th chi-squar distribution. LO 15-3 Comput a goodnss-o-it tst or unqual xpctd rquncis. LO 15-4 Conduct a tst o hypothsis to vriy that data groupd into a rquncy distribution ar a sampl rom a normal distribution. LO 15-5 Us graphical and statistical mthods to dtrmin whthr a st o sampl data is rom a normal distribution. LO 15-6 Prorm a chi-squar tst or indpndnc on a contingncy tabl. 15-

3 LO 15- List and xplain th charactristics o th chi-squar distribution. Charactristics o th Chi-Squar Distribution Th major charactristics o th chi-squar distribution: Positivly skwd. Non-ngativ. Basd on dgrs o rdom. Whn th dgrs o rdom chang, a nw distribution is cratd. 15-3

4 LO 15-1 Conduct a tst o hypothsis comparing an obsrvd st o rquncis to an xpctd distribution. Goodnss-o-Fit Tst: Comparing an Obsrvd St o Frquncis to an Expctd Distribution Lt 0 and b th obsrvd and xpctd rquncis, rspctivly. Hypothss: H 0 : Thr is no dirnc btwn th obsrvd and xpctd rquncis. H 1 : Thr is a dirnc btwn th obsrvd and th xpctd rquncis. 15-4

5 Goodnss-o-it Tst: Comparing an Obsrvd St o Frquncis to an Expctd Distribution LO 15-1 Th tst statistic is: o Th critical valu is a chi-squar valu with (k 1) dgrs o rdom, whr k is th numbr o catgoris. 15-5

Bor doing so, h dcids to hir Magnolia Rsarch, LLC to conduct a survy o adults about thir avorit mal whn ating out.

6 LO 15-1 Goodnss-o-Fit Exampl Bubba, th ownr o Bubba s Fish and Pasta, a chain o rstaurants locatd along th Gul Coast o Florida, is considring adding stak to his mnu. Bor doing so, h dcids to hir Magnolia Rsarch, LLC to conduct a survy o adults about thir avorit mal whn ating out. Magnolia slctd a sampl 10 adults and askd ach to indicat thir avorit mal whn dining out. Th rsults ar rportd in th tabl. Is it rasonabl to conclud thr is no prrnc among th our ntrés? 15-6

7 LO 15-1 Goodnss-o-Fit Exampl Stp 1: Stat th null hypothsis and th altrnat hypothsis. H 0 : Thr is no dirnc btwn o and. H 1 : Thr is a dirnc btwn o and. Stp : Slct th lvl o signiicanc. α = 0.05 as statd in th problm. Stp 3: Slct th tst statistic. Th tst statistic ollows th chi-squar distribution, dsignatd as χ. o 15-7

Goodnss-o-Fit Exampl Stp 4: Formulat th dcision rul. 7.

8 Goodnss-o-Fit Exampl Stp 4: Formulat th dcision rul i Rjct H.05,3 1.05,4 1, 1, 0 o o o k o k LO

9 LO 15-1 Goodnss-o-Fit Exampl Critical Valu 15-9

10 LO 15-1 Goodnss-o-Fit Exampl Stp 5: Comput th valu o th chi-squar statistic and mak a dcision. o 15-10

LO 15-1 Goodnss-o-Fit Exampl 7.815 Critical Valu.0 Th computd χ o.0 is lss than th critical valu o 7.815. Th dcision, thror, is to ail to rjct H 0 at th.

11 LO 15-1 Goodnss-o-Fit Exampl Critical Valu.0 Th computd χ o.0 is lss than th critical valu o Th dcision, thror, is to ail to rjct H 0 at th.05 lvl. Conclusion: Th dirnc btwn th obsrvd and th xpctd rquncis is du to chanc. Thr is no dirnc in prrnc toward th our ntrés

12 LO 15-1 Chi-squar MgaStat 15-1

13 LO 15-3 Comput a goodnss-o-it tst or unqual xpctd rquncis. Goodnss-o-Fit Tst: Unqual Expctd Frquncis Lt 0 and b th obsrvd and xpctd rquncis, rspctivly. Hypothss: H 0 : Thr is no dirnc btwn th obsrvd and xpctd rquncis. H 1 : Thr is a dirnc btwn th obsrvd and th xpctd rquncis

14 LO 15-3 Goodnss-o-Fit Tst: Unqual Expctd Frquncis Exampl Th Amrican Hospital Administrators Association (AHAA) rports th ollowing inormation concrning th numbr o tims snior citizns ar admittd to a hospital during a on-yar priod. Forty prcnt ar not admittd; 30 prcnt ar admittd onc; 0 prcnt ar admittd twic, and th rmaining 10 prcnt ar admittd thr or mor tims. A survy o 150 rsidnts o Bartow Estats, a community dvotd to activ sniors locatd in cntral Florida, rvald 55 rsidnts wr not admittd during th last yar, 50 wr admittd to a hospital onc, 3 wr admittd twic, and th rst o thos in th survy wr admittd thr or mor tims. Can w conclud th survy at Bartow Estats is consistnt with th inormation suggstd by th AHAA? Us th.05 signiicanc lvl

15 Goodnss-o-Fit Tst: Unqual Expctd Frquncis Exampl Stp 1: Stat th null hypothsis and th altrnat hypothsis. H 0 : Thr is no dirnc btwn local and national xprinc or hospital admissions. H 1 : Thr is a dirnc btwn local and national xprinc or hospital admissions. Stp : Slct th lvl o signiicanc. α = 0.05 as statd in th problm. Stp 3: Slct th tst statistic. Th tst statistic ollows th chi-squar distribution, dsignatd as χ. LO

16 Goodnss-o-Fit Tst: Unqual Expctd Frquncis Exampl Stp 4: Formulat th dcision rul i Rjct H.05,3 1.05,4 1, 1, 0 o o o k o k LO

17 LO 15-3 Goodnss-o-Fit Tst: Unqual Expctd Frquncis Exampl Distribution statd in th problm Frquncis obsrvd in a sampl o 150 Bartow rsidnts Expctd rquncis o sampl i th distribution statd in th null hypothsis is corrct Computation o 0.40 X 150 = X 150 = X 150 = X 150=

18 Goodnss-o-Fit Tst: Unqual Expctd Frquncis Exampl Stp 5: Comput th valu o th Chi-squar statistic and mak a dcision LO 15-3 o Computd χ 15-18

19 LO 15-3 Goodnss-o-Fit Tst: Unqual Expctd Frquncis Exampl Th computd χ o is in th Do not rjct H 0 rgion. Th dirnc btwn th obsrvd and th xpctd rquncis is du to chanc. W conclud that thr is no vidnc o a dirnc btwn th local and national xprinc or hospital admissions

Tsting th Hypothsis that a Distribution o Data Is rom a Normal Population Rcall th rquncy

20 LO 15-4 Conduct a tst o hypothsis to vriy that data groupd into a rquncy distribution is a sampl rom a normal distribution. Tsting th Hypothsis that a Distribution o Data Is rom a Normal Population Rcall th rquncy distribution o Applwood s proits rom th sal o 180 vhicls. Th rquncy distribution is rpatd blow. Is it rasonabl to conclud that th proit data is a sampl obtaind rom a normal population? 15-0

21 Tsting th Hypothsis that a Distribution o Data Is rom a Normal Population LO 15-4 Stp 1: Calculat th probabilitis or ach class. Convrt ach class limit into a z- scor using a man o $1, and a standard dviation o $643.63, thn ind th probability. 15-1

22 Tsting th Hypothsis that a Distribution o Data Is rom a Normal Population Stp : Us ths probabilitis to comput th xpctd rquncis or ach class X 180 = 3.85 LO

23 Tsting th Hypothsis that a Distribution o Data Is rom a Normal Population LO 15-4 Stp 3: Comput th chi-squar statistic using: 15-3

24 Tsting th Hypothsis that a Distribution o Data Is rom a Normal Population LO 15-4 Stp 4: Compar th computd statistic to th critical statistic and mak a statistical conclusion: H 0 : Th population o proits ollows th normal distribution H 1 : Th population o proits dos not ollow th normal distribution 15-4

Graphical Approach to Conirm Normality: Andrson-Darling Tst LO 15-5 Us graphical mthods to dtrmin whthr a st o sampl data is rom a normal distribution. Stp 1: Crat cumulativ distributions. a. Cumulativ distribution o th raw data.

25 Graphical Approach to Conirm Normality: Andrson-Darling Tst LO 15-5 Us graphical mthods to dtrmin whthr a st o sampl data is rom a normal distribution. Stp 1: Crat cumulativ distributions. a. Cumulativ distribution o th raw data. b. Cumulativ normal distribution. Stp : Compar th cumulativ distributions. a. Sarch th largst absolut numrical dirnc btwn th distributions. b. Using a statistical tst, i th dirnc is larg, thn w rjct th null hypothsis that th data is normally distributd. Th rd dots in th graph rprsnt th proit o ach o th 180 vhicls rom th Applwood Auto Group, and th blu lin, which is mostly covrd by th rd dots, rprsnts a normal cumulativ distribution. Th graph shows that th proit data closly ollows th blu lin and that th distribution o proits ollows a normal distribution rathr closly. 15-5

26 Contingncy Tabl Analysis A contingncy tabl is usd to invstigat whthr two traits or charactristics ar rlatd. Each obsrvation is classiid according to two critria. W us th usual hypothsis tsting procdur. Th dgrs o rdom is qual to: LO 15-6 Prorm a chi-squar tst or indpndnc on a contingncy tabl. (numbr o rows 1)(numbr o columns 1). Th xpctd rquncy is computd as: 15-6

27 LO 15-6 Contingncy Analysis W can us th chi-squar statistic to ormally tst or a rlationship btwn two nominal-scald variabls. To put it anothr way, Is on variabl indpndnt o th othr? Ford Motor Company runs an assmbly plant in Darborn, Michigan. Th plant oprats thr shits pr day, 5 days a wk. Th quality control managr wishs to compar th quality lvl on th thr shits. Vhicls ar classiid by quality lvl (accptabl, unaccptabl) and shit (day, atrnoon, night). Is thr a dirnc in th quality lvl on th thr shits? That is, is th quality o th product rlatd to th shit whn it was manuacturd? Or is th quality o th product indpndnt o th shit on which it was manuacturd? A sampl o 100 drivrs, who wr stoppd or spding violations, was classiid by gndr and whthr or not th drivrs wr waring a satblt whn stoppd. For this sampl, is waring a satblt rlatd to gndr? Dos a mal rlasd rom dral prison mak a dirnt adjustmnt to civilian li i h rturns to his homtown or i h gos lswhr to liv? Th two variabls ar adjustmnt to civilian li and plac o rsidnc. Not that both variabls ar masurd on th nominal scal. 15-7

28 LO 15-6 Contingncy Analysis Exampl Th Fdral Corrction Agncy is invstigating th qustion, Dos a mal rlasd rom dral prison mak a dirnt adjustmnt to civilian li i h rturns to his homtown or i h gos lswhr to liv? To put it anothr way, is thr a rlationship btwn adjustmnt to civilian li and plac o rsidnc atr rlas rom prison? Us th.01 signiicanc lvl. 15-8

good, air, or unsatisactory. Th classiications or th 00 ormr prisonrs wr tallid as ollows.

29 Contingncy Analysis Exampl LO 15-6 Th agncy s psychologists intrviwd 00 randomly slctd ormr prisonrs. Using a sris o qustions, th psychologists classiid th adjustmnt o ach individual to civilian li as outstanding, good, air, or unsatisactory. Th classiications or th 00 ormr prisonrs wr tallid as ollows. Josph Camdn, or xampl, rturnd to his homtown and has shown outstanding adjustmnt to civilian li. His cas is on o th 7 tallis in th uppr lt box (circld). 15-9

30 Contingncy Analysis Exampl LO 15-6 Stp 1: Stat th null hypothsis and th altrnat hypothsis. H 0 : Thr is no rlationship btwn adjustmnt to civilian li and whr th individual livs atr bing rlasd rom prison. H 1 : Thr is a rlationship btwn adjustmnt to civilian li and whr th individual livs atr bing rlasd rom prison. Stp : Slct th lvl o signiicanc. α = 0.01 as statd in th problm. Stp 3: Slct th tst statistic. Th tst statistic ollows th chi-squar distribution, dsignatd as χ

31 Contingncy Analysis Exampl Stp 4: Formulat th dcision rul i Rjct H.01,3.01,(1)(3) 1) 1)( 4,( 1) 1)(,( 0 o o o o c r LO

32 Computing Expctd Frquncis ( ) LO 15-6 (10)(50)

33 LO 15-6 Computing th Chi-squar Statistic 15-33

34 LO 15-6 Conclusion 5.79 Th computd χ o 5.79 is in th Do not rjct H 0 rgion. Th null hypothsis is not rjctd at th.01 signiicanc lvl. W conclud thr is no vidnc o a rlationship btwn adjustmnt to civilian li and whr th prisonr rsids atr bing rlasd rom prison. For th Fdral Corrction Agncy s advismnt program, adjustmnt to civilian li is not rlatd to whr th x-prisonr livs

35 LO 15-6 Contingncy Analysis Minitab 15-35

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points