11.3 Goodness-of-fit test

In this type of hypothesis test, you determine whether the data "fit" a particular distribution or not. For example, you may suspect your unknown data fit a binomialdistribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. The null and the alternate hypotheses for this test may be written in sentences or may be stated as equations orinequalities.

The test statistic for a goodness-of-fit test is:

where:

• $O$ = observed values (data)
• $E$ = expected values (from theory)
• $k$ = the number of different data cells or categories

The observed values are the data values and the expected values are the values you would expect to get if the null hypothesis were true. There are $n$ terms of the form $\frac{(O-E{)}^2}{E}$ .

The degrees of freedom are $\text{df = (number of categories - 1)}$ .

The goodness-of-fit test is almost always right tailed. If the observed values and the corresponding expected values are not close to each other, then the test statisticcan get very large and will be way out in the right tail of the chi-square curve.

Employers particularly want to know which days of the week employees are absent in a five day work week. Most employers wouldlike to believe that employees are absent equally during the week. Suppose a random sample of 60 managers were asked on which day of the week did they have the highest number of employee absences. The results were distributed as follows: