Categorical Goodness of Fit Test

1. Categorical Goodness of Fit Test

2. Hypothesis Test – Goodness of Fit Kerpuztin’s Syndrome A random sample of cases of Kerpuztin’s Syndrome (KS) is collected, and the severity of each case is noted as: (F)atal, (S)evere, (Mo)derate, or (Mi)ld. The case severities are listed below: S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, F, F, F, F, F, F, F Test the following: Null: The severity levels are distributed as: 40% Severe, 30% Moderate, 20% Mild and 10% Fatal against the Alternative: The severity levels are not distributed this way. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries. Fully discuss the testing procedure and results. This discussion must include a clear discussion of the population and the null hypothesis, the family of samples, the family of errors and the interpretation of the p- value.

3. Test the following: Null: The severity levels are distributed as: 40% Severe, 30% Moderate, 20% Mild and 10% Fatal against the Alternative: The severity levels are not distributed this way. Show your work. Completely discuss and interpret your test results, as indicated in class and case study summaries. Fully discuss the testing procedure and results. This discussion must include a clear discussion of the population and the null hypothesis, the family of samples, the family of errors and the interpretation of the p-value.

4. Observed Sample Counts Severe: 15 observed S, S, S, S, S, S, S, S, S, S, S, S, S, S, S Moderate: 12 observed Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mo, Mild: 16 observed Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi, Mi Fatal: 7 observed F, F, F, F, F, F, F

5. Fatal Observed 7 Expected .10*50=5 Error Term ((7-5)2)/5 = .80

6. Severe Observed 15 Expected .40*50=20 Error Term ((15-20)2)/20 = 1.25

7. Moderate Observed 12 Expected .30*50=15 Error Term ((12-15)2)/15= .60

8. Mild Observed 16 Expected .20*50=10 Error Term ((16-10)2)/10 = 3.6

9. Total Error  6.26 on 4 categories 4 6.2514 0.100 p  .100 = 10.0% Categories p-value Total Error

10. Total Error Categories ERROR p-value 4 0.0000 1.000 4 0.5844 0.900 4 1.0052 0.800 4 1.4237 0.700 4 1.8692 0.600 4 2.3660 0.500 4 2.6430 0.450 4 2.9462 0.400 4 3.2831 0.350 4 3.6649 0.300 4 4.1083 0.250 4 4.6416 0.200 4 4.9566 0.175 4 5.3170 0.150 4 5.7394 0.125 46.25140.100 4 6.4915 0.090 4 6.7587 0.080 4 7.0603 0.070 4 7.4069 0.060 4 7.8147 0.050 4 8.3112 0.040 4 8.9473 0.030 4 9.8374 0.020 p-value p-value ≈ .10 or 10% Category Count

11. Our population of interest consists of cases of Kerpuztin’s Syndrome (KS). Each member of our Family of Samples is a single random sample of 50 cases of KS. Our Family of Samples consists of all such samples. An error is obtained from each member of the FoS as: Error = (nmild – emild)2/(emild) + (nmoderate – emoderate)2/(emoderate) + (nsevere – esevere)2/(esevere) + (nfatal – efatal)2/(efatal) Obtaining errors of this type from every member of the FoS yields the Family of Errors. If the true severity proportions for the population of KS cases are 40% Severe, 30% Moderate, 20% Mild and 10% Fatal, then approximately 10.0% of the Family of Samples yield errors as bad as or worse than our sample. We do not reject the null hypothesis.