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12-2 Chapter Outline 12.1Chi-Square Goodness of Fit Tests 12.2A Chi-Square Test for Independence

12-3 12.1 Chi-Square Goodness of Fit Tests 1.Carry out n identical trials with k possible outcomes of each trial 2.Probabilities are denoted p 1, p 2, …, p k where p 1 + p 2 + … + p k = 1 3.The trials are independent 4.The results are observed frequencies, f 1, f 2, …, f k

12-4 Chi-Square Goodness of Fit Tests Continued Consider the outcome of a multinomial experiment where each of n randomly selected items is classified into one of k groups Let f i = number of items classified into group i (i th observed frequency) E i = np i = expected number in i th group if p i is probability of being in group i (i th expected frequency)

12-5 A Goodness of Fit Test for Multinomial Probabilities H 0 : multinomial probabilities are p 1, p 2, …, p k H a : at least one of the probabilities differs from p 1, p 2, …, p k Test statistic: Reject H 0 if  2 >   2 or p-value <   2 and the p-value are based on p-1 degrees of freedom

12-6 Example 12.1: The Microwave Oven Preference Case Tables 12.1 and 12.2

12-7 Example 12.1: The Microwave Oven Preference Case Continued

12-8 Example 12.1: The Microwave Oven Preference Case #3 Figure 12.1

12-9 A Goodness of Fit Test for Multinomial Probabilities f i =the number of items classified into group i E i =np i H 0 :The values of the multinomial probabilities are p 1, p 2,…p k H 1 :At least one of the multinomial probabilities is not equal to the value stated in H 0

12-10 A Goodness of Fit Test for a Normal Distribution Have seen many statistical methods based on the assumption of a normal distribution Can check the validity of this assumption using frequency distributions, stem-and-leaf displays, histograms, and normal plots Another approach is to use a chi-square goodness of fit test

12-11 A Goodness of Fit Test for a Normal Distribution Continued 1.H 0 : the population has a normal distribution 2.Select random sample 3.Define k intervals for the test 4.Record observed frequencies 5.Calculate the expected frequencies 6.Calculate the chi-square statistics 7.Make a decision

12-12 12.2 A Chi-Square Test for Independence Each of n randomly selected items is classified on two dimensions into a contingency table with r rows an c columns and let f ij = observed cell frequency for i th row and j th column r i = i th row total c j = j th column total Expected cell frequency for i th row and j th column under independence

12-13 A Chi-Square Test for Independence Continued H 0 : the two classifications are statistically independent H a : the two classifications are statistically dependent Test statistic Reject H 0 if  2 >   2 or if p-value <    2 and the p-value are based on (r-1)(c-1) degrees of freedom

12-14 Example 12.3 The Client Satisfaction Case Table 12.4

12-15 Example 12.3 The Client Satisfaction Case #2 Figure 12.2 (a)

12-16 Example 12.3 The Client Satisfaction Case #3 H 0 : fund time and level of client satisfaction are independent H 1 : fund time and level of client satisfaction are dependent Calculate frequencies under independence assumption Calculate test statistic of 46.44 Reject H 0