137
HYPOTHESIS-TESTING HYPOTHESIS-TESTING DR. C. P. GUPTA DR. C. P. GUPTA MANAGEMENT DEVELOPMENT INSTITUTE MANAGEMENT DEVELOPMENT INSTITUTE GURGAON GURGAON

Hypothesis - Testing

Embed Size (px)

Citation preview

HYPOTHESIS-TESTING DR. C. P. GUPTA MANAGEMENT DEVELOPMENT INSTITUTEGURGAONFirst some basics ? HYPOTHESIS ??? isformallystatedexpectationabouthowa behaviour operates. isapropositionthataresearcher wants to verify. isaconjecturalstatementofthe relation between two or more variables. BASIC ELEMENTS OF A TEST OF HYPOTHESISFormulate a Null Hypothesis (H0). Formulate an Alternative Hypothesis (H1) Select a suitable Test Statistic Specify a Level of Significance (o) Define a suitable Decision Criterion based on o and Test Statistic Make necessary Assumptions if required Experiment and Calculation of Test Statistic Conclusion or DecisionP - Value .???? ProbabilityValueorp-valueistheprobabilityofobservingasample outcomeevenmoreextremethantheobservedvaluewhenthenull hypothesis is true. Thesmallerthep-value,thesmallerarethechancesthatvariationsare caused by chance/random factors. It is also called observed level of significance. Itprovidesanalternativewaytodecidewhetheranullhypothesisistobe accepted. It has following advantages and thats the reason mostly statistical softwares are giving printouts with p - values: it allows a decision maker to use his/her own level of significance and makedecisionaccordinglyoncesampleresultsareavailablewith necessary statistic itprovidesverypreciseinformationaboutthehighestlevelof significance at which the null hypothesis must be accepted. So far, we were preparing ourselves for the necessary backdrop to take forward a solid move for understanding hypothesis-testing. For every hypothesis-testing problem, we require a test which may be PARAMETRICorNON PARAMETRIC PARAMETRIC TEST ... A PARAMETRIC test is a test whose model requires and specifies certain conditions about the parameters of the population from which the sample is drawn. Such tests makes certain assumptions about the nature of the underlying population like Normal Probability Distribution and their validity rests upon the validity of these assumptions. These test are more powerful and strong in their assertions and are usually applicable when data is interval scale or Ratio Scale. These tests are very much rich and developed. NON PARAMETRIC TESTS... These tests are also known as distribution free methods. These are the tests whose model does not specify conditions and assumptions about the parameters of the population; they lack parameters. These are widely used for nominal or ordinal data where no parametric tests are not available at all. However, they can also be used for Ratio or Interval Scale data as well. NON PARAMETRIC TESTS (continued) ... These tests are not very powerful and strong in their assertions. Non-parametric statistical tests are typically much easier to learn and apply than are parametric tests. These tests usually convert data into ranks (hence, such tests are also sometimes known as Rank Tests) or signs and thereby may loose some important information. First, we take HYPOTHESIS-TESTING RELATEDTONOMINAL DATA .And, use NON-PARAMETRIC TESTS ONE SAMPLE NOMINAL DATA Can we conclude from the above data that there is a significant difference between those who say YES and those who say NO? Do you think this event (Commonwealth Games) is an opportunity toshowcase rich Indian cultural heritage to world?Yes/NoNO YESCount280270260250240230220LOOKING FOR A PROPER TEST NOMINAL DATA ONE SAMPLE ONE SAMPLE AND POPULATION HAS ONLY TWO CATEGORIES: THE BINOMIAL TEST APPLICABLETOPOPULATIONSHAVINGONLYTWO CLASSES THEPROPORTIONOFCLASSONEWOULDBEpAND THAT OF ANOTHER CLASS 1 - p NULLHYPOTHESISIS-WHETHERTHEPOPULATION PROPORTION IS p IFTHESAMPLESIZEISSMALL-APPLYBINOMIAL PROBABILITY DISTRIBUTION. IFTHESAMPLESIZEISLARGE-APPLYNORMALPROBABILITY DISTRIBUTION. Before starting or afterthe journey were you or will you stay in hotel / Dharmshala etc. YES NOWhat is the thing you wish to PUT ON TEST? UNDERSTANDING SPSS OUTPUTBinomial TestYES 268 .54 .50 .118aNO 232 .46500 1.00Group 1Group 2TotalDo you think this event(Commonweal th Games)i s an opportuni ty toshowcase ri ch Indi ancul turalheri tage to worl d?Yes/NoCategory NObservedProp. Test Prop.Asymp. Si g.(2-tai l ed)Based on Z Approxi mati on.a. Can we conclude forthe above that people watch different types of sports equally? LOOKING FOR A PROPER TEST 2. Which of the following sports do you like most to watch on TV a. Athletics b. Cricket c. Hockey d. WWF e. Any other (please specify) __________________ Cases weighted by FREQSports you like most to watch on TVOthers WWF Hockey Cricket AthleticsCount100806040200NOMINAL DATA ONE SAMPLE (CONTINUED) ONE SAMPLE AND POPULATION HAS MANY CATEGORIES: - THE CHI- SQUARE (_2) TEST APPLICABLETOPOPULATIONSHAVINGONLYTWO OR MORE CLASSES THEOBJECTIVEOFTHISISTOTESTWHETHERTHE DISTRIBUTIONOFOBSERVATIONSINVARIOUS CATEGORIESISACCORDINGTOSOMEEXPECTED PATTERN. NULLHYPOTHESISIS-WHETHERTHEOBSERVED DISTRIBUTIONISASPEREXPECTEDDISTRIBUTION. i.e. A TEST OF GOODNESS OF FIT. Mode of transport from your destination railway station to home/office/hotel. Own Vehicle Taxi Bus OthersWhat is the thing you wish to PUT ON TEST? Test Statistics48.6544.000Chi-SquareadfAsymp. Si g.Sports youl i ke most towatch on TV0 cell s (.0%) have expected frequenci es l ess than5. The mi nimum expected cel l frequency i s 52.0.a. UNDERSTANDING SPSS OUTPUTNPar Tests Chi-Square Test Sports you like most to watch on TV50 52.0 -2.080 52.0 28.070 52.0 18.015 52.0 -37.045 52.0 -7.0260Athl eti csCri cketHockeyWWFOthersTotalObserved N Expected N Resi dualTWO SAMPLES NOMINAL DATA Now, about Related SAMPLES!!!! To appreciate, the concept of RELATED SAMPLE, lets look at... Is there any statistically significant difference in the state of ENO before and after joining a health club? Is there any statistically significant difference in the state of the person before and after taking coffee? Related Samples... .aredefinedasthosewheretheobservationsin onehassomerelationorinfluenceonthoseofthe other sample. How do we visualize in SPSS? Research Project: IMPACT OF PLAY SCHOOL EDUCATION ON THE PERSONALITY OF CHILDREN One of the research issue in it - whether children joining play school become more independent and confident What to do ? NOMINAL DATA TWO SAMPLES (RELATED) - THE Mc NEMAR TEST APPLICABLEtocaseswhereresearchdesignisBEFORE AND AFTER; usually, it is used to test the effectiveness of a treatment conducted on one set of respondentsApplicable for TWO SAMPLES WITH TWO CLASSES To apply it, one has to set up a table of the following format: -+ + AB -CD AANDDSHOWCHANGESBETWEENRESPONSES.IFTHE TREATMENTHASNOIMPACT,THENHALFOF(A+D)MUST CHANGEINONEDIRECTIONWHILEOTHERHALFSHOULD CHANGE IN OTHER DIRECTION. NOMINAL DATA TWO SAMPLES (RELATED) (CONTINUED) - THE Mc NEMAR TEST NULLHYPOTHESISIS-WHETHERTHE PROBABILITYOFA,P(A),ISEQUALTO PROBABILITY OF D, P(D); i.e. P(A) = P(D) = 1/2. FORIT,WEHAVE _2=(A-D)2/(A+D)WITHdf=1 WITHTHECORRECTIONFORCONTINUITY- _2 = (|A-D|-1)2/(A+D)WITH df = 1 IFTHEEXPECTEDFREQUENCYISSMALLER THAN 5, THEN USE THE BINOMIAL TEST. Were you satisfied with the billed amounts prior to privatization? YES NO Were you satisfied with the billed amounts after privatization? YES NO What is the thing you wish to PUT ON TEST? What would you like to test in such a case? LOOKING FOR A PROPER TEST 1. Before the present mess contractor, was the food oily and/or Spicy? YES/NO 2. Is the food of the present mess contractor oily and/or Spicy?YES/NO Before the present mess contractor, was the food oily and/or Spicy?NO YESCount11010090807075100Is the food of the present mess contractor oily and/or Spicy?NO YESCount1101009080706010570Test Statisticsb1759.344.002NChi-SquareaAsymp. Si g.Before the present messcontractor, was the food oil y and/orSpicy? & Is the food of the presentmess contractor oi ly and/or Spicy?Conti nui ty Correcteda. McNemar Testb. UNDERSTANDING SPSS OUTPUTMcNemar Test Before the present mess contractor, was the foodoily and/or Spicy? & Is the food of the present messcontractor oily and/or Spicy?40 6030 45Before the present messcontractor, was the foodoi ly and/or Spi cy?121 2Is the food of the presentmess contractor oil yand/or Spi cy?What would you like to test in such a case? LOOKING FOR A PROPER TEST Research Project: HEALTH OF THE STUDENTS IN THE INDIAN SCHOOL One of the research issue in it - How do students perceive their health? and the researcher is interested in knowing whether this perception is different among male and female students. NOMINAL DATA TWO SAMPLES (UNRELATED) - THE CHI - SQUARE TEST APPLICABLE IN CASES HAVING TWO SAMPLES WITH K CLASSES. ITCANBEUSEDEVENINBEFORE&AFTER SITUATIONS PROVIDED IT HAS K CLASSES. IT FOLLOWS CHI-SQUAREDISTRIBUTIONWITHdf = (K-1). NOMINAL DATA TWO SAMPLES (UNRELATED) (CONTINUED) THE FISHER EXACT PROBABILITY TEST applicable when the data has only two unrelated data; and the sample size is small (preferably less than 20) To apply it, one has to set up a table of the following format: Group xGroup y Group I AB Group II CD theexactprobabilityofobservingaparticularsetof frequenciesina2X2tableisgivenbythe HYPERGEOMETRIC DISTRIBUTION - NOMINAL DATA TWO SAMPLES (UNRELATED) (CONTINUED) On this basis and referring to necessary Table, one can decide whether H0 is to be accepted or rejected. ||.|

\|+||.|

\|+||.|

\|+=B ANBD BAC Ap From the following data, can we conclude whether a particular Fund is performing better than the other? What is the thing you wish to PUT ON TEST?

PERFORMING BETTER THAN MARKET PERFORMING WORSE THAN MARKETSECTOR FUNDS2515 BALANCED FUNDS2225 SPSS OUTPUT FOR FINAL DECISION DECISION ????? Crosstabs FUNDS * PERFORMANCE OF FUNDS CrosstabulationCount25 15 4022 25 4747 40 87SECTOR FUNDSBALANCED FUNDSFUNDSTotalPERFORMING BETTERTHAN MARKETPERFORMING WORSETHAN MARKETPERFORMANCE OF FUNDSTotalChi-Square Tests2.142b1 .1431.557 1 .2122.155 1 .142.196 .1062.118 1 .14687Pearson Chi -SquareConti nui ty CorrectionaLi kel i hood Rati oFi sher's Exact TestLi near-by-Li nearAssoci ati onN of Val i d CasesValue dfAsymp. Si g.(2-si ded)Exact Si g.(2-si ded)Exact Si g.(1-si ded)Computed onl y for a 2x2 tablea. 0 cell s (.0%) have expected count l ess than 5. The mi ni mum expected count i s18.39.b. MORE THAN TWO SAMPLES NOMINAL DATA Research Project: QUALITY OF MANAGEMENT IN PUBLIC INSTITUTIONS One of the research issue in it - Do public have different experiences in dealing with public institutions like DDA, MCD, NDMC, etc.? What to do ? NOMINAL DATA MORE THAN TWO SAMPLES (RELATED) - THE COCHRAN Q TESTITISANEXTENSIONOFMcNEMARTESTFORK- RELATED SAMPLES. COCHRANQTESTISUSEDTOTESTWHETHER THREEORMORERELATEDSETSOFFREQUENCIES DIFFER SIGNIFICANTLY AMONG THEMSELVES. ITISAPPLICABLEWHENRESPONSESAREOF DICHOTOMOUSINNATURE-YESORNO; SUCCESSFUL OR UNSUCCESSFUL. NOMINAL DATA MORE THAN TWO SAMPLES (RELATED)(CONTINUED) - THE COCHRAN Q TESTTHE FORMULA IS - WHERE K= NUMBER OF COLUMNS; Ri= TOTAL OF ith ROW; Cj= TOTAL OF jth COLUMN; S =SUM OF TOTAL SCORES; C =MEAN OF COLUMNS TOTAL. =i2iR S Kj2C) -j(C ) 1 K ( KQ Assume that five members A, B, C, D, and E of a mountaineering club each attempt three different rock climb at each of which they either succeed or fail. The outcomes are shown below read 0 as fail and 1 as success. MEMBERSABC DE CLIMB#111001 CLIMB#210010 CLIMB#301111 What is the thing you wish to PUT ON TEST? SPSS OUTPUT FOR FINAL DECISION DECISION ????? NPar Tests Cochran Test Frequencies2 33 21 4CLIMB#1CLIMB#2CLIMB#30 1ValueTest Statistics51.200a2.549NCochran's QdfAsymp. Si g.1 is treated as a success.a. Research Project: IMPACT OF Sarve Shiksha Abhiyan ON THE RURAL DEVELOPMENT IN INDIA One of the research issue in it- Whether no. of children from different castes in villages going to primary schools, middle schools and high schools are different. What to do ? NOMINAL DATA MORE THAN TWO SAMPLES (UNRELATED) - THE CHI - SQUARE TESTIt is an extension of 2 independent samples. Thenullhypothesisiswhetherthereexistsa significantdifferencebetweentheK independent groups. Assume that you want to judge the Financial Analysts ability to predict correctly share prices in the market. For that you collected the following data for 100 days about the prediction of 5 analysts about a particular share. What is the thing you wish to PUT ON TEST? FORECAST WITHIN ACCEPTABLE RANGE BEYOND ACCEPTABLE RANGE ANALYSTS A3565 B4555 C3664 D4852 E5050 SPSS OUTPUT FOR FINAL DECISION DECISION ????? Crosstabs FINANCIAL ANALYSTS * FORECAST OF SHARE PRICES CrosstabulationCount35 65 10045 55 10036 64 10048 52 10050 50 100214 286 500ABCDEFINANCIALANALYSTSTotalWITHIN ACCEPTABLERANGEBEYOND ACCEPTABLERANGEFORECAST OF SHARE PRICESTotalChi-Square Tests7.794a4 .0997.843 4 .0974.439 1 .035500Pearson Chi -SquareLi kel i hood Rati oLi near-by-Li nearAssoci ationN of Val id CasesValue dfAsymp. Si g.(2-si ded)0 cell s (.0%) have expected count l ess than 5. Themini mum expected count i s 42.80.a. SUMMARY OF TESTS RELATED TO NOMINAL DATA ONE SAMPLE ONE SAMPLE NOMINAL DATA 2 CATEGORIESMORE THAN 2 CATEGORIES BINOMIAL TEST _2 SUMMARY OF TESTS RELATED TO NOMINAL DATA TWO SAMPLES TWO SAMPLES NOMINAL DATA RELATED SAMPLESUNRELATEDSAMPLES 2 CLASSES _2 MORE THAN 2 CLASSES McNemar Test _2 SUMMARY OF TESTS RELATED TO NOMINAL DATA MORE THAN TWO SAMPLES MORE THAN 2 SAMPLES NOMINAL DATA RELATED SAMPLESUNRELATEDSAMPLES 2 CLASSES _2 MORE THAN 2 CLASSES Cochran Q Test _2 AFTER TESTING FOR NOMINAL DATA; NOW WE MOVE TO HYPOTHESIS-TESTING RELATEDTOORDINAL DATA .ONE SAMPLE ORDINAL DATA Research Project: PUBLIC SCHOOLS IN DELHI A STUDY One of the research issue in it - Do public have different preferences in sending their wards to private public school, government-aided schools and government schools? What to do ? What would you like to test in such a case? LOOKING FOR A PROPER TEST Research Project: Satisfaction Survey among participants of National Management Program at Management Development Institute, Gurgaon One of the research issue in it - LOOKING FOR A PROPER TEST Campus (location, layout, etc.)and hostel facilities (room, , etc.)Highly SatisfiedModerately SatisfiedNeither DissatisfiedModerately DissatisfHighly DissatisfiedCount8070605040307560355040ORDINAL DATA ONE SAMPLE - THEKOLMOGOROV-SMIRNOVTEST(K-S TEST ) K - S one sample test is a test of goodness of fitfor ordinal data. Ittestswhetherthereexistsanydifferenceinthe distributionofobservedvaluesandtheexpectedvalues according to some specified distribution. It is based on the concept of cumulative frequency. Ifboththedistributionsareidenticalthenthedeviations amongthemwouldbeapproximatelyzero;otherwise deviations would be very large. ORDINAL DATA ONE SAMPLE (continued) - THE KOLMOGOROV - SMIRNOV TEST ( K-S TEST ) For One Tailed Test - D = MAXIMUMSn1(X)- Sn(X)where, Sn1=Proportion of Cum. Frequency of distribution one; Sn =Proportion of Cum. Frequency of theoretical distribution; For Two Tailed Test,takeD = MAXIMUM( Sn1(X)- Sn(X) )that is with sign. If sample size is large, then one should use the followingformula: _2= 4 D2 (n1 n2)/( n1+ n2 ) with df = 2. What according to you should be the ideal period of review? a) 1 Month b) 2 Months c) 3 Months d) 4 Months What is the thing you wish to PUT ON TEST? SPSS OUTPUT FOR FINAL DECISION DECISION ????? One-Sample Kolmogorov-Smirnov Test210 2221 124 65.229 .531.205 .054-.229 -.5313.312 7.910.000 .000NMini mumMaximumUniform Parametersa,bAbsol utePosi ti veNegati veMost ExtremeDi fferencesKolmogorov-Smi rnov ZAsymp. Si g. (2-tai l ed)REVIEWPERIOD FREQ_1Test di stri buti on i s Uni form.a. Calculated from data.b. NPar Tests Research Project: DETERMINANTS OF MOTIVATIONAL LEVEL OF THE FLOOR-LEVEL WORKERS A CASE STUDY OF PUNCHKULA PLANT One of the research issue in it - Is the motivation level of floor works at PUNCHKULA PLANT higher than that of the industry? What to do ? ORDINAL DATA ONE SAMPLE (continued) - THE SIGN TEST FOR THE MEDIAN Itisusedtotestwhetherthemedianvalueofthesampleis same as that of the population, i.e. Ho:MEDIAN = Mo IfH1 : MEDIAN = Mo; then T = MIN(n1,n2); IfH1 : MEDIAN < Mo; then T = n1; IfH1 : MEDIAN > Mo; then T = n2; where n1 = Number of observations < Mo; and, n2 = Number of observations > Mo; For large samples, one may use Binomial Distribution with p = 1/2andn=n 1+n 2orNormalDistribution,with=1n/2AND o = 1 \ n /2 Celebrity increases appeal of a product - 1. Strongly Disagree 2. Disagree 3. Neither Disagree nor Agree 4. Agree 5. Strongly Agree What is the thing you wish to PUT ON TEST? SPSS OUTPUT FOR FINAL DECISION DECISION ????? NPar Tests Test Statisticsa3203.0081.5691.00079.0781.000NMedi anChi-SquaredfAsymp. Si g.Chi-SquaredfAsymp. Si g.Yates' Conti nui tyCorrecti onCELEBRITYINCREASESPRODUCTAPPEALGroupi ng Variabl e: VAR00001a. TWO SAMPLES ORDINAL DATA (RELATED) Research Project: A STUDY OF EFFECTIVENESS OF TRAINING PROGRAMMES CONDUCTED BY NSE FOR SHARE BROKERS AND SUB-BROKERS IN INDIA One of the research issue in it - Is the comfort level of brokers/sub-brokers in trading mechanism increased after training? What to do ? ORDINAL DATA TWO SAMPLES(RELATED) - THE SIGN TEST It is applicable for two samples which are related. Ittestswhetherthereexistsanydifferencebetweentheobservationsoftworelated samples. NULL HYPOTHESIS : P(A < B) = P(A>B) = 1/2; i.e. if there isnodifferenceintherelatedobservationsthennumberof changes of higher values over other must be equal to number of changes of lower values over other. ForSmallSample:USEBINOMIALDISTRIBUTION ANDforlargesamples:USENORMALDISTRIBUTION with MEAN = (1/2)N AND VARIANCE = 1/4 N where N = total number of signs. ORDINAL DATA TWO SAMPLES(RELATED)(continued) - THE SIGN TEST ToapplytheSignTest,ourdatashouldbepresentedasfollowsand workoutthesignofdifferenceamongthevaluesofSampleAand Sample B. Sample A Sample B Sign123::ORDINAL DATA TWO SAMPLES (RELATED) - THE WILCOXON MATCHED PAIRS SIGNED RANK TEST It is applicable for two samples which are matched. SIGN TEST tells the direction but not the magnitude. Init,thedifferencesarefoundandrankedignoringthe sign.(Droptheobservationwithzerodifferenceand assign average rank in case of a tie). NULLHYPOTHESIS:Noimpactoftreatment.Hence, thesumofpositiveranksmustbeequaltothatof negative ranks. FORONETAILTEST-T+orT-dependingupon ALTERNATIVE HYPOTHESIS.FOR TWO - TAIL TEST - MIN(T+ ,T-). ORDINAL DATA TWO SAMPLES (RELATED) (CONTINUED) - THE WILCOXON MATCHED PAIRS SIGNED RANK TEST For small sample, USE t STATISTICS and for large samplesuseNORMALPROBABILITY DISTRIBUTIONwithMEAN=(N(N+1)/4)ANDVARIANCE=(N(N+1)(2N+1)/24)whereN= Total Numbers of Pairs less dropped outs. Do you and your boss go the joint goal setting meeting after doing proper groundwork based on MOU targets for the department? (Tick one) YOUYOUR BOSS a) Always b) Often c) Sometimes d) Rarely e) Never What is the thing you wish to PUT ON TEST? SPSS OUTPUT FOR FINAL DECISION DECISION ????? NPar Tests Frequencies7977924041993Negati ve DifferencesaPosi ti ve Di fferencesbTi escTotalYOUR BOSS - YOUNYOUR BOSS < YOUa. YOUR BOSS > YOUb. YOUR BOSS = YOUc. Sign Test Test Statisticsa-.100.920ZAsymp. Si g. (2-tai l ed)YOUR BOSS- YOUSi gn Testa. SPSS OUTPUT FOR FINAL DECISION DECISION ????? NPar Tests Wilcoxon Signed Ranks Test Ranks797a774.35 617153.50792b815.78 646101.50404c1993Negati ve RanksPosi ti ve RanksTi esTotalYOUR BOSS - YOUN Mean Rank Sum of RanksYOUR BOSS < YOUa. YOUR BOSS > YOUb. YOUR BOSS = YOUc. Test Statisticsb-.802a.423ZAsymp. Si g. (2-tai l ed)YOUR BOSS- YOUBased on negati ve ranks.a. Wil coxon Si gned Ranks Testb. TWO SAMPLES ORDINAL DATA (UNRELATED) Research Project: QUALITY OF MANAGEMENT IN INDIAN CORPORATE SECTOR One of the research issue in it - Is there any significant difference in the opinions of shareholders and the management about the quality of governance in India? What to do ? Research Project: PHYSICAL FITNESS AMONG STUDENTS OF INDIA One of the research issue in it - Is there any significant difference in PHYSICAL FITNESS of boy students and girl students? What to do ? Another Problem ORDINAL DATA TWO SAMPLES (UNRELATED) - THE MEDIAN TEST It is applicable for two samples which are independent. It is a test of central tendency. Ho : M1= M2 For this test, find the grand median and prepare a table as follows: SAMPLE 1 SAMPLE 2 ABOVE GRAND MEDIANABA+B BELOW GRAND MEDIANCDC+D A+CB+DN ORDINAL DATA TWO SAMPLES (UNRELATED) (continued) - THE MEDIAN TEST It follows _2 distribution where with df = 1 N(AD - BC- N/ 2 )2_2 =(A+B) (A+C) (B+C) (B+D)ORDINAL DATA TWO SAMPLES (UNRELATED) - THEMANNWHITNEYTEST(alsoknownasWilcoxonRank Sum Test ) It is applicable for two samples which are independent. It is a test of difference of central tendency of two samples. It is an extension of t -Test. It ranks the observations of both samples as if they are coming from the same population. And, then define U-Statistic as thus - U1 = (n1 n2 ) + (n1(n1+1)/2) - Ranks of n1 where n1 is size of the sample with fewer observations. U2 = (n1 n2 ) - U1;and U = Minimum(U1, U2) Forlargesamples,itfollowsnormalprobabilitydistributionwithmean= (n1n2)/2 and variance = (n1n2)( n1 + n2 +1)/12 Your mobile is Pre-paid(1) or Post-paid(2) Monthly usage/bill in Rs. (Mark at appropriate option) a) 2000 What is the thing you wish to PUT ON TEST? SPSS OUTPUT FOR FINAL DECISION DECISION ????? NPar Tests Ranks467 481.94 225065.50517 502.04 259554.50984MOBILE...?PREPAIDPOST PAIDTotalMONTHLYUSAGE/BILL IN Rs.N Mean Rank Sum of RanksMann-Whitney Test Test Statisticsa115787.500225065.500-1.124.261Mann-Whi tney UWi l coxon WZAsymp. Si g. (2-tai l ed)MONTHLYUSAGE/BILLIN Rs.Groupi ng Vari able: MOBILE...?a. SPSS OUTPUT FOR FINAL DECISION DECISION ????? NPar Tests Mann-Whitney Test Ranks12 26.75 321.0036 23.75 855.0048GROUPING VARIABLEBOYSGIRLSTotalPHYSICALFITNESS SCOREN Mean Rank Sum of RanksTest Statisticsa189.000855.000-.643.520Mann-Whi tney UWi l coxon WZAsymp. Si g. (2-tai l ed)PHYSICALFITNESSSCOREGroupi ng Vari able: GROUPING VARIABLEa. There is no difference between median spending. Do you believe that there is no difference between the spending of Pre- and Post-Mobile Users? USAGE DISTRIBUTION BEHAVIOUR AMONGPRE- AND POST-PAID MOBILE USERSMONTHLY USAGE/BILL>20001000-2000700-1000500-700300-500