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 with estimated population parameters is pre- cisely as in Section 14.1, except that the number of degrees of freedom for the chi-square random variable is degrees of freedom = (K-m-1) (14.3) where K is the number of categories and m is the number of unknown popula- tion parameters. A Test for the Poisson Distribution Consider a test to determine if data are generated by the Poisson distribution. One pro- cedure for attempting to resolve questions of disputed authorship is to count the num- ber of occurrences of particular words in blocks of text. These can be compared with results from passages whose authorship is known; often this comparison can be achieved through the assumption that the number of occurrences follows a Poisson distribution. An example of this type of research involves the study of The Federalist Papers (Mosteller and Wallace 1964). x3 Example 14.4 Federalist Papers (Chi-Square) For a sample of 262 blocks of text (each approximately 200 words in length) from The Fed- eralist Papers (Mosteller and Wallace 1964), the mean number of occurrences of the word may was 0.66. Table 14.7 shows the observed frequencies of occurrence of this word in the 262 sampled blocks of text. Test the null hypothesis that the population distribution of oc- currences is Poisson, without assuming prior knowledge of the mean of this distribution