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Chapter 12

Chapter 12. Chi-Square Tests. Chapter Outline. 12.1 Chi-Square Goodness of Fit Tests 12.2 A Chi-Square Test for Independence. 12.1 Chi-Square Goodness of Fit Tests. Carry out n identical trials with k possible outcomes of each trial

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Chapter 12

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  1. Chapter 12 Chi-Square Tests

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

  3. 12.1 Chi-Square Goodness of Fit Tests • Carry out n identical trials with k possible outcomes of each trial • Probabilities are denoted p1, p2, … , pkwhere p1 + p2 + … + pk= 1 • The trials are independent • The results are observed frequencies, f1, f2, …, fk

  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 fi = number of items classified into group i (ith observed frequency) • Ei = npi = expected number in ith group if pi is probability of being in group i (ith expected frequency)

  5. A Goodness of Fit Test for Multinomial Probabilities • H0: multinomial probabilities are p1, p2, … , pk • Ha: at least one of the probabilities differs from p1, p2, … , pk • Test statistic: • Reject H0 if • 2 > 2 or p-value <  • 2 and the p-value are based on p-1 degrees of freedom

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

  7. Example 12.1: The Microwave Oven Preference Case Continued

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

  9. A Goodness of Fit Test for Multinomial Probabilities fi = the number of items classified into group i Ei =npi H0: The values of the multinomial probabilities are p1, p2,…pk H1: At least one of the multinomial probabilities is not equal to the value stated in H0

  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

  11. A Goodness of Fit Test for a Normal Distribution Continued • H0: the population has a normal distribution • Select random sample • Define k intervals for the test • Record observed frequencies • Calculate the expected frequencies • Calculate the chi-square statistics • Make a decision

  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 • fij = observed cell frequency for ith row and jth column • ri = ith row totalcj = jth column total • Expected cell frequency for ith row and jth column under independence

  13. A Chi-Square Test for Independence Continued • H0: the two classifications are statistically independent • Ha: the two classifications are statistically dependent • Test statistic • Reject H0 if 2 > 2 or if p-value <  • 2 and the p-value are based on (r-1)(c-1) degrees of freedom

  14. Example 12.3 The Client Satisfaction Case Table 12.4

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

  16. Example 12.3 The Client Satisfaction Case #3 • H0: fund time and level of client satisfaction are independentH1: fund time and level of client satisfaction are dependent • Calculate frequencies under independence assumption • Calculate test statistic of 46.44 • Reject H0

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