Goodness of Fit Tests Qualitative (Nominal) Data Counts of occurrences for each category (fi) Null Hypothesis Something that yields predicted number of expected count for each category (ei) Test Statistic 2 = (fi-ei)2 / ei Decision Rule Reject H0 if 2 is “large” (2 > some 2 critical value)

Specific skewness depends on degrees of freedom 2 Distribution Specific skewness depends on degrees of freedom 5 d.f 10 d.f 15 d.f

2 Distribution 10 d.f P(2 > ) = 0.05 P(2 < ) = 0.95 ?

2 Distribution Critical Values Degrees of Freedom Area in Upper Tail 0.10 0.05 0.01 0.001 1 2.706 3.841 6.635 10.827 2 4.605 5.991 9.210 13.815 3 6.251 7.815 11.345 16.266 4 7.779 9.488 13.277 18.466 5 9.236 11.070 15.086 20.515 6 10.645 12.592 16.812 22.457 7 12.017 14.067 18.475 24.321 8 13.362 15.507 20.090 26.124 9 14.684 16.919 21.666 27.877 10 15.987 18.307 23.209 29.588

Equality of Percentages Example An HR manager wants to know whether 30% of the workers are very satisfied, 30% are somewhat satisfied, 20% are indifferent, and 20% are quite unhappy with their jobs. It is your job to determine the answer, but you only have the resources to question 300 workers. What do you tell the HR manager (assume a 5% level of significance)? From a random survey of 300 workers, you tabulate the following: Number of workers very satisfied 76 somewhat satisfied 100 indifferent 76 unhappy 48

Independent Events Example You want to know whether race is related to the shift that the worker is working. An intern sample 228 workers in the factory and classifies them as follows: Shift 1 Shift 2 Shift 3 White 42 28 16 Non-white 44 57 41 Can you conclude that race and shift are related (i.e., not independent)? (use  = 0.05)