Goodness-of-Fit Tests and Contingency Tables: Chi-Square Random Variable

1. Chapter 14 Goodness-of-Fit Tests and Contingency Tables

2. The Chi-Square Random Variable A random sample of n observations each of which can be classified into exactly one of K categories is selected. Denote the observed numbers in each category by O1, O2, . . ., OK. If the null hypothesis (H0) specifies probabilities 1, 2, . . ., K for an observation falling into each of these categories, the expected numbers in the categories under H0 would be If the null hypothesis in true and the sample size is large enough so that the expected values are at least five, then the random variable associated with has, to a good approximation, a chi-square distribution with (K – 1) degrees of freedom.

3. A Goodness-of-Fit Test A goodness-of-fit test, of significance level , of H0 against the alternative that the specified probabilities are not correct is based on the decision rule where 2 K-1,  is the number for which And the random variable 2 K-1 follows a chi-square distribution with (K – 1) degrees of freedom.

4. Goodness-of-Fit Tests When Population Parameters are Estimated Suppose that a null hypothesis specifies category probabilities that depend on the estimation (from the data) of m unknown population parameters. The appropriate goodness-of-fit test of the null hypothesis when population parameters are estimated is the same as that previously mentioned, except that the number of degrees of freedom for the chi-square random variable is Where K is the number of categories.

5. Bowman-Shelton Test for Normality The Bowman-Shelton Test for Normality is based on the closeness to 0 of the sample skewness and the closeness to 3 of the sample kurtosis. The test statistic is It is known that as the number of sample observations becomes very large, this statistic has, under the null hypothesis that the population distribution is normal, a chi-square distribution with 2 degrees of freedom. The null hypothesis is, of course, rejected for large values of the test statistic.

6. r x c Contingency Table(Table 14.8)

7. Chi-Square Random Variable for Contingency Tables It can be shown that under the null hypothesis the random variable associated with has, to a good approximation, a chi-square distribution with (r – 1)(c – 1) degrees of freedom. The approximation works well if each of the estimated expected numbers Eij is at least 5. Sometimes adjacent classes can be combined in order to meet this assumption.

8. A Test of Association in Contingency Tables Suppose that a sample of n observations is cross classified according to two attributes in an r x c contingency table. Denote by Oij the number of observations in the cell that is in the ith row and the jth column. If the null hypothesis is The estimated expected number of observations in this cell, under H0, is Where Ri and Cj are the corresponding row and column totals. A test of association at a significance level  is based on the following decision rule

9. Bowman-Shelton Test for Normality 2 Random Variable Goodness-of-Fit Tests Specified Parameters Unknown Parameters Poisson Distribution Normal Distribution Kurtosis Skewness Test of Association Key Words