A Goodness-of-Fit Test
This article explains how to make tests of hypothesis about experiments with more than two possible outcomes (or categories). Such experiments, called multinomial experiments, possess four characteristics. Note that a binomial experiment is a special case of a multinomial experiment. An experiment with the following characteristics is called a multinomial experiment.
1. It consists of n identical trials (repetitions).
2. Each trial results in one of k possible outcomes (or categories), where k _ 2.
3. The trials are independent.
4. The probabilities of the various outcomes remain constant for each trial.
An experiment of many rolls of a die is an example of a multinomial experiment. It consists of many identical rolls (trials); each roll (trial) results in one of the six possible outcomes; each roll is independent of the other rolls; and the probabilities of the six outcomes remain constant for each roll. As a second example of a multinomial experiment, suppose we select a random sample of people and ask them whether or not the quality of American cars is better than that of Japanese cars. The response of a person can be yes, no, or does not know. Each person included in the sample can be considered as one trial (repetition) of the experiment. There will be as many trials for this experiment as the number of persons selected. Each person can belong to any of the three categories—yes, no, or does not know. The response of each selected person is independent of the responses of other persons. Given that the population is large, the probabilities of a person belonging to the three categories remain the same for each trial. Consequently, this is an example of a multinomial experiment.
The frequencies obtained from the actual performance of an experiment are called the observed frequencies. In a goodness-of-fit test, we test the null hypothesis that the observed frequencies for an experiment follow a certain pattern or theoretical distribution. The test is
1. It consists of n identical trials (repetitions).
2. Each trial results in one of k possible outcomes (or categories), where k _ 2.
3. The trials are independent.
4. The probabilities of the various outcomes remain constant for each trial.
An experiment of many rolls of a die is an example of a multinomial experiment. It consists of many identical rolls (trials); each roll (trial) results in one of the six possible outcomes; each roll is independent of the other rolls; and the probabilities of the six outcomes remain constant for each roll. As a second example of a multinomial experiment, suppose we select a random sample of people and ask them whether or not the quality of American cars is better than that of Japanese cars. The response of a person can be yes, no, or does not know. Each person included in the sample can be considered as one trial (repetition) of the experiment. There will be as many trials for this experiment as the number of persons selected. Each person can belong to any of the three categories—yes, no, or does not know. The response of each selected person is independent of the responses of other persons. Given that the population is large, the probabilities of a person belonging to the three categories remain the same for each trial. Consequently, this is an example of a multinomial experiment.
The frequencies obtained from the actual performance of an experiment are called the observed frequencies. In a goodness-of-fit test, we test the null hypothesis that the observed frequencies for an experiment follow a certain pattern or theoretical distribution. The test is