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# APPLIED STATISTICS STATS 0110B

UCLA

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This 5 page Class Notes was uploaded by Isobel Stanton on Friday September 4, 2015. The Class Notes belongs to STATS 0110B at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 108 views. For similar materials see /class/177961/stats-0110b-university-of-california-los-angeles in Statistics at University of California - Los Angeles.

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Date Created: 09/04/15

UCLA STAT 110B Applied Statistics for Engineering and the Sciences Categorical Data olnstructor IVO Dinov Asst Prof In Statistics and Neurology uTeaching Assistants Brian Ng UCLA Statistics University of California Los Angeles Spring 2003 httpwwwstaLuclaedudinovcoursesstudentshtml mum r rm 11m my 1 Categorical Data is that which counts the number of outcomes falling into various categories Binomial Experiment 7 consists of two categories Multinomial Experiment 7 consist of more than two categories Binomial Experiment 0 n independent trials 0 Two possible outcomes S success and F failure 0 p Probability of success on each trial 0 X Number of successes in n trials Binomial Distribution Pdf EX VarX Multinomial Experiment n independent trials results in one of k possible categories labeled 1 k pi the probability of a trial resulting in the ith category where p1 pk l 0 Ni number of trials resulting in the ith category where N1 Nk n Multinomial Cont d The random variables Np Nk have a multinornial distribution pn1nk P1 mp 11 nit41k Multinomial Cont d 0 Expected Value EN HP E Variance Var Ni npiqi o Covariance Cov Np Nj 39npipj Testing Goodness of Fit with Specified Cell Probabilities We wish to test whether the cell probabilities are specified by p1 pk where p1 pk 1 We will use a test statistic to compare the observed cell count Ni to the expected cell count under Ho E nPi Hup1 p1 and and pk pk Ha Some pi 5 pi Test Statistic k 2 X2 2 Ni Er i1 E f This is a Pearson s goodnessoffit statistic Rejection Region X2 gt x0 where x2 is the chisquared distribution with k1 degrees of freedom General Rule We want npi 2 5 for all cells a A Example A study is run to see whether the public favors the construction of a new dam It is thought that 40 favor dam construction 30 are neutral 39 39 A39 gt r r i and the rest have not thought about it A random sample of 150 individuals are interviewed resulting in 42 in favor 61 neutral 33 opposed and the rest have not though about it Does the data indicate that the stated proportions are incorrect Use oc001 Example Cont d Ho p104 p203 p302 p401 Ha At least one probability is not as specified Test Statistic Xz Rejection Region X2gt X20 0173 1134 Favor Neutral 0 ose Unaware Total 42 61 3 14 150 7 427602 617 452 337302 60 45 30 2 14715 15 X2 1146 Since X2 1146 gt 31001372 1134 we reject Ho Conclude that at least one of the true proportions differs from that hypothesized u A Goodness of Fit for Distributions Continuous and Discrete 0 Uses the concept of Maximum Likelihood Estimations MLE The range of a hypothesized distribution is divided into a set of k intervals cells After finding the MILE of unknown parameters the cell probabilities are calculated and the x2 test performed Found in many computer packages SOCR Testing Normality Many test procedures that we have developed rely on the assumption of Normality There are many test for Normality of data One uses the normal to provide cell probabilities for the chisquare goodnessoffit test A better test is based on the Normal Probability Plot Testing Normality Cont d Recall The NPP should be approx linear for normal data and the correlation coefficient is a measure of linearity lfr is much less than one we would conclude that the data doesn t come from a Normal distribution RyanJoiner Test 1 Order the data X1Xn 2 Compute the normal percentiles 1 i 375 I n 25 3 Compute the correlation coefficient R for the yixi pairs and look up the distribution table for the RyanJoiner Statistics Al 2 m 31 RyanJoiner Test 4 State the Null and Alternative Hypotheses Ho The population is normal Hz The population is not normal 5 Specify alpha and obtain critical values from Table Al2 Compare R to this value Example I Consider the following data Use the Ryan oiner test to test the assumption of norrnalit at on 010 E R 115413412913612612214 a Dal12911413213412613613613 128145129128138155146132 Normal01 random sample Example m m mI39I mEE Ascendm H Datals Normal Mama N 01 1 Data 0999999999 OANwhmmummo CorrN01Data095541 Testing Homogeneity of Populations We wish to compare I multinomial populations each with I categories Take ni samples from the ith population Let Nij be the number of observations from the i3911 population in the j3911 category Hence 2139 Nu i Place the data in a I X I table Corresponding to each cell there is a cell probability pfprobability and outcome for the i3911 population falls into the j3911 category where 21 pij 1 Table Category 1 Total 1 n11 n12 n1J n1 2 n21 n22 n2J n2 Pop i nl1 n12 nlJ ril Total n1 n2 nJ n 11 Test Ho P1j sz Prj gtJ Ir Ha Some pij 5 pr estimated by Under Ho the common cell probability pj is Category 1 2 1 p11 p12 p1J 2 p21 p22 p2J Pop I p1 p12 PIJ 77 A Test Cont d l The estimated expected cell frequency is A nn E 7113 l 39J l l J J n The test statistic is X2 quot1 Evy EU Rejection Region X2 gt xza with df 11 I l 24 A Testing for Association Individuals are categorized by two categorical variables We wish to determine w ether these variables are associated Row Categories 7 AlAI Column Categories 7 B1BJ n Total number of observations nij the number of individuals classified as Ai an BJ Hence 22 nij n Ho PAimBJ PAiPBJ for all ij Ha Some PAi Bj 5 PAiPBJ Expected Frequency A Ili X IL i39 J 11 Test Statistic X2 2 2 quotv Evy EU Rejection Region X2 gt XZGL with d f 11 11 2 A The Chisquare distribution Lotto after 399 numbers have been drawn Do some numbers appear more frequently in LOTTO 51 I 00 13 a X 7 12 a 15 901 In 6 u 1 12 m 139 n n 150 16 a 1701 I 1 w 5 2n 1 2 7 22 g n 2 2A 12 2s a 26 4 27 m m z 29 14 an 12 3 on 32 12 n m 3 n 35 a 36 3 31 14 3100 3 a w d m Number on ball Freguency ofLOTTO Winning numbers 2a A Total number ofballs selected 577399 Expected value of each number 39940 9975 Observed x2 statistics is x03097 df40 139 Pvalue 0817 Conclusion No evidence for departure from the null hypothesis

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