12.10: (Student CD-ROM Topic) Chi-Square Goodness-of-Fit Tests
(STUDENT CD-ROM TOPIC) CHI-SQUARE GOODNESSOF-FIT TESTS
In this section, χ2 goodness-of-fit tests are used to determine whether a set of data matches a specific probability distribution. Goodness-of-fit tests compare the observed frequencies in a category to the frequencies that are theoretically expected if the data follow a specific probability distribution. To perform a χ2 goodness-of-fit test, you follow several steps. First, you determine the specific probability distribution to compare to the data. Second, you estimate (from a sample) or hypothesize the value of each parameter (such as the mean) of the selected probability distribution. Next, you determine the theoretical probability in each category using the selected probability distribution. Finally, you use a χ2-test statistic to test whether the selected distribution is a good fit to the data.
Chi-Square Goodness-of-Fit Test for a Poisson Distribution
Recall from Section 5.4 that you used the Poisson distribution to find the probability of a specific number of arrivals per minute at a bank located in the central business district of a city. Suppose that you recorded the actual arrivals per minute in 200 one-minute periods over the course of a week. Table 12.22 summarizes the results.
TABLE 12.22 Frequency Distribution of Arrivals per Minute
Arrivals 0 1 2 3 4 5 6 7 8
Frequency 14 31 47 41 29 21 10 5 002 200
To determine whether the number of arrivals per minute follows a Poisson distribution, the null and alternative hypotheses are: H0: The number of arrivals per minute follows a Poisson distribution. H1: The number of arrivals per minute does not follow a Poisson distribution. The Poisson distribution has one parameter, its mean λ, and you need to specify its value in the null and alternative hypotheses. You can use either a λ. value based...