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Chapter 12

Chapter 12. Chi-Square Tests. Chi-Square Tests. 12.1 Chi-Square Goodness of Fit Tests 12.2 A Chi-Square Test for Independence. The Multinomial Experiment. 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. Chi-Square Tests 12.1 Chi-Square Goodness of Fit Tests 12.2 A Chi-Square Test for Independence

  3. The Multinomial Experiment • 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 of the number of trials that result in each of k possible outcomes, denoted f1, f2, …, fk

  4. Chi-Square Goodness of Fit Tests • 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 OvenPreference Case • H0: p1 = .20, p2 = .35, p3 = .30, p4 = .15 • Ha: H0 fails to hold

  7. Example 12.1: The Microwave OvenPreference Case Continued

  8. 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

  9. Example 12.2: The Car Mileage Case • Consider the 50 gas mileage samples from Chapter 1 (Table 1.4) • A histogram is symmetrical and bell-shaped • This suggests a normal distribution • Will test this using a chi-square goodness of fit test

  10. Example 12.2: The Car Mileage Case #2 • First divide the number line into intervals • Will use the class boundaries of the histogram in Figure 2.10 • Table below gives intervals and observed frequencies

  11. Example 12.2: The Car Mileage Case #3 • Next step is to calculate expected frequencies • Those calculations are shown below

  12. Example 12.2: The Car Mileage Case #4 • Will test: • H0: Population is normally distributed • Ha: Population is not normally distributed • 2=0.43242 < 20.05=11.0705 cannot reject H0

  13. 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

  14. 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

  15. Example 12.3: The Client SatisfactionCase • A financial institution sells investment products • Examining whether customer satisfaction depend on the type of product purchased • Data shown in Table 12.4

  16. Example 12.3: The MegaStat Output

  17. Example 12.3: The MINITAB Output

  18. Example 12.3: The Client SatisfactionCase H0: Client satisfaction is independent of fund type Ha: Client satisfaction depends upon fund type

  19. Example 12.3: Plots of Row Percentages Versus Investment Type

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