[教育]張俊妮概率統(tǒng)計(jì)北大光華-lecture27goodnessoffitandcontingencytabl

1、1,Lecture 23,Test of Goodness of FitContingency Tables,Test of Goodness-of-fitExample: Scott Marketing Research firm,Over the past year market shares have stabilized, with 30% for company A, 50% for company B, 20% for

2、 company C. Recently company C has developed a “new and improved” product that will replace its current entry in the market. Managers of company C have asked Scott firm to determine whether the new product will ca

3、use a shift in the market shares of the three competitors.,Scott firm conducted a survey with 200 responses. The results are:A 48B 98C 54Is there any change in market shares?,Goodness-of-fit Test,H0: Th

4、e population follows a multinomial distribution with PA=0.30 PB=0.50 PC=0.20； H1: The population does not follow a multinomial distribution with PA=0.30 PB=0.50 PC=0.20.Basic idea： if H0 is true, the differenc

5、e between the observed frequencies and expected frequencies should not be large.,Sampling Distribution under H0,,: the number of categories.,: the sample size.,: the observed counts for category i.,Rule of five,The te

6、st statistic used to perform the test is only approximately Chi-squared distributed. For the approximation to apply, the expected cell frequency has to be at least 5.If the expected frequency in a cell is less than 5

7、, combine it with other cells.,Calculations: Scott Marketing Research firm,Assumed Observed ExpectedProportion Frequency Frequency Company A 0.3 48

8、60 Company B 0.5 98 100 Company C 0.2 54 40 Total 200 200 Using ?= 0.05，rejection region is,,,,,,,5.99,Any change in market shares？Yes!,Genetics Problem,Each individual

9、 in a given population must have one of three possible genotypes, and it is assumed that the probabilities p1, p2 and p3 of the three genotypes can be represented in the following form:A random sample is taken from t

10、he population, and the statistician must test the following hypotheses:H0: There exists a value of q in (0,1) such that pi=pi(q) for i=1,…,3,H1: The hypothesis H0 is not true.,Let denote the M.L.E. of the para

11、meter based on the observed numbers N1,…,N3, i.e.,The statistic Q can be found by:R. A. Fisher (1924) showed that if the null hypothesis is true and certain regularity conditions are satisfied, then the d.f. of Q co

12、nverges to the d.f. of as the sample size where s is the number of parameters estimated (s=1 in the genetic example).,Contingency Tables,Alber’s Brewery manufactures and distributes three type

13、s of beer: light, regular, and dark. In an analysis of the market segments for the three beers, the firm’s market research group has raised the question of whether the preferences for the three beers differ among male a

14、nd female beer drinkers. If the preference depends on the gender of the beer drinker, the firm will tailor its promotions to different target markets.,H0: Beer preference is independent of the gender of the beer drinker

15、.H1: Beer preference is not independent of the gender of the beer drinker.Sample results:,R rows and C columns.pij: the probability that an individual selected at random from a given population will be classified in

16、the ith row and the jth column of the table.pi.: the marginal probability that the individual will be classified in the ith row of the table.p.j: the marginal probability that the individual will be classified in the

17、 jth column of the table.,Nij: the number of individuals who are classified in the ith row and the jth column of the table.Ni.: the number of individuals who are classified in the ith row of the table.N.j: the number

18、 of individuals who are classified in the jth column of the table.,The Test of Independence,H0: pij=pi.p.j for i=1,…,R and j=1,…,CH1: The hypothesis H0 is not true.Notice that under the null hypothesis, only pi

19、. and p.j need to be estimated, and the number of such parameters is (R-1)+(C-1)=R+C-2.Based on the observed counts, Under H0,,Under H0,,The Beer Example,Expected Frequency:,When ? = 0.05, the rejection region is

20、So we reject the null hypothesis and conclude beer preferences is not independent of the sex of the beer drinker.,Example: Montana Outlook Poll,The Bureau of Business and Economic Research at the University of Montana

21、conducted a poll of opinions of Montana residents in May 1992. Participants were asked about their income range and whether their personal financial status was worse, the same, or better than one year ago.,Example Conti