578 CHAPTER 11 Goodness-of-Fit and Contingency Tables Key Concept By “goodness-of-fit” we mean that sample data consisting of observed frequency counts arranged in a single row or column (called a one-way frequency table) agree with some particular distribution (such as normal or uniform) being considered. We will use a hypothesis test for the claim that the observed frequency counts agree with the claimed distribution. 11-1 Goodness-of-Fit DEFINITION A goodness-of-fit test is used to test the hypothesis that an observed frequency distribution fits (or conforms to) some claimed distribution. Testing for Goodness-of-Fit Objective Conduct a goodness-of-fit test, which is a hypothesis test to determine whether a single row (or column) of frequency counts agrees with some specific distribution (such as uniform or normal). Notation O represents the observed frequency of an outcome, found from the sample data. E represents the expected frequency of an outcome, found by assuming that the distribution is as claimed. k represents the number of different categories or cells. n represents the total number of trials (or the total of observed sample values). p represents the probability that a sample value falls within a particular category. Requirements 1. The data have been randomly selected. 2. The sample data consist of frequency counts for each of the different categories. 3. For each category, the expected frequency is at least 5. (The expected frequency for a category is the frequency that would occur if the data actually have the distribution that is being claimed. There is no requirement that the observed frequency for each category must be at least 5.) Null and Alternative Hypotheses H0: The frequency counts agree with the claimed distribution. H1: The frequency counts do not agree with the claimed distribution. Test Statistic for Goodness-of-Fit Tests x2 = a1O - E22 E P-values: P-values are typically provided by technology, or a range of P-values can be found from Table A-4. Critical values: 1. Critical values are found in Table A-4 by using k - 1 degrees of freedom, where k is the number of categories. 2. Goodness-of-fit hypothesis tests are always right-tailed. KEY ELEMENTS
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