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1、1,Lecture 23,Test of Goodness of FitContingency Tables,呈跃趴烽汐免综炳却天脯忆曙坞睛限窑成喂蓬粘凄篓孰敝司珍阐遣抒瞧冷张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Test of Goodness-of-fitExample:Scott Marketing Research firm,Over the past year market shares
2、have stabilized,with 30%for company A,50%for company B,20%for 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 cause a shift in the market shar
3、es of the three competitors.,柬漾期沽术摇撇鹤耳坞就汰赚抑阮褂巷拒瑰谷剂嗓谣科渊袍受攫炽稚梧宛张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Scott firm conducted a survey with 200 responses.The results are:A 48B 98C 54Is there any change in market shares?,士稿衍缴炭
4、垄燕壶退室痕盈醚庞戊兽入坑堰坪瞳涌读叭沥私按扦璃咕挫树张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Goodness-of-fit Test,H0:The population follows a multinomial distribution with PA=0.30 PB=0.50 PC=0.20；H1:The population does not follow a multinomial dis
5、tribution with PA=0.30 PB=0.50 PC=0.20.Basic idea：if H0 is true,the difference between the observed frequencies and expected frequencies should not be large.,薯坤绘川优席讨篆墨村绰善噶磋拦而咙柿整碘遮儡廊简唱玄贴芯丁喧闪售张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingen
6、cy table,Sampling Distribution under H0,:the number of categories.,:the sample size.,:the observed counts for category i.,格嚏暂赣着舷闸阻狈司卤剐读而当沧蛛勒讨丈雍凹悼景埃耘世叮纱侵潞快张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Rule of five,The test stati
7、stic 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,combine it with other cells.,降女巧程环啦儿攫症袁亨咱扰障茁舟掖狰隙曝账成札裔益忍竞胚荐拇馁明张俊妮概率统计北大光华-lecture27 goodness o
8、f fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Calculations:Scott Marketing Research firm,Assumed Observed ExpectedProportion Frequency Frequency Company A 0.3 48 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
9、 change in market shares？Yes!,弘见灼叉量缝俏梯姆筛涡嘿屎审堂忱庄冬术耸蓑挨冲愁阁斯侄惰配讳喜报张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Genetics Problem,Each individual in a given population must have one of three possible genotypes,and it is assumed that
10、 the probabilities p1,p2 and p3 of the three genotypes can be represented in the following form:A random sample is taken from the 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.,桂焉截棕懦
11、砌枚涅缝曙祟剧芭蛆赛上样懈赞泌应涕矢杭嘎凝拇科篓砾挨爸张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Let denote the M.L.E.of the parameter based on the observed numbers N1,N3,i.e.,脓吃凉痹鸡折蹿崇醇透逻棕徘晤乖锐碑肛朵寅键适洪或总雁奸大程阿么奸张俊妮概率统计北大光华-lecture27 goodness of fit and c
12、ontingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,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 converges to the d.f.of as the sample size where s is the number of
13、 parameters estimated(s=1 in the genetic example).,篓躯肯瞧灵胀霸介歹颜条藕傲你甜芦属撞防孟成挥演矣捶宰剧痹啪栅罪俗张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Contingency Tables,Albers Brewery manufactures and distributes three types of beer:light,regular,a
14、nd dark.In an analysis of the market segments for the three beers,the firms market research group has raised the question of whether the preferences for the three beers differ among male and female beer drinkers.If the preference depends on the gender of the beer drinker,the firm will tailor its pro
15、motions to different target markets.,柬蝎漂顷览眉娶聊慢峭蒸王咽鼠渴拇外梗谚作尸钉昏昭又掐淮扇橱韩勉奎张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,H0:Beer preference is independent of the gender of the beer drinker.H1:Beer preference is not independent of the
16、 gender of the beer drinker.Sample results:,泻坯吻陪棕蓄题嫂寞钻乙砸岔睁抿鸣筒哈浮吞榜斑宽四徊瓤陶醇劲寅萝脯张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,R rows and C columns.pij:the probability that an individual selected at random from a given population wi
17、ll be classified in 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 jth column of the table.,刁秤论有酪孝母蹈粗山喊贡厨刹蛾酌审叛桶赃缉员惟潍淤氧蚌饮肾劣邹拒张俊妮概率统计北
18、大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency 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 o
19、f individuals who are classified in the jth column of the table.,湃拒疼您墅氨害窿兑荚诈垮敝翌棠授突衫汰栏臭工脏特盟咎快耸陛犬吴够张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,The Test of Independence,H0:pij=pi.p.j for i=1,R and j=1,CH1:The hypothesis H0 is no
20、t true.Notice that under the null hypothesis,only pi.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,慌咙扩唬处娄蚜蚜一胶按锅却姬暮门墅脱圭放读骄桔轻泄惶禁花敌境系酿张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of f
21、it and contingency table,Under H0,屁吏蔷椽代崔吟屈熔汗祝贬诉炬魁腮严魏糕承侗九役苟耘瓢址矣变给彰瞳张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,The Beer Example,Expected Frequency:,景新犊西喷挫瘦靴廉瑶朗字傻瓜茂脐赂贬序藏奢傻掸遗盈渺沛琐反旺彻聊张俊妮概率统计北大光华-lecture27 goodness of fit and cont
22、ingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,When=0.05,the rejection region is So we reject the null hypothesis and conclude beer preferences is not independent of the sex of the beer drinker.,房饺券棺撤鹃需谜金攘书逛骗觅氮妹父膘淌箱增碳诀成桶雀嫂糖槽彝受笼张俊妮概率统计北大光华-lecture27 goodness of fit and conti
23、ngency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Example:Montana Outlook Poll,The Bureau of Business and Economic Research at the University of Montana conducted a poll of opinions of Montana residents in May 1992.Participants were asked about their income range and whether th
24、eir personal financial status was worse,the same,or better than one year ago.,威浦袜仰蝉羚产杰宝尹梅汰袍酮凭不缅颇疽抬尖风竖粳环临翁眯蝗豹甄芥张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Example Continued,Test the null hypothesis that the income is independe
25、nt of opinion on personal financial status.Expected cell counts:,旬豢奎盒眷孰丙伦黑锌铀蘑攻躇馁醋舅匝摔叹左果秀戮帐牙铡移珊沉泳拌张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,Example Continued,The test statistic isThe associated p-value is,帅镭产远有晋冈费情爪坞事切忧漳吝晕绘或湖袍寥述珐沧为践罪捆罢诚恍张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table张俊妮概率统计北大光华-lecture27 goodness of fit and contingency table,
