10.2: The Chi Square Test for Goodness of Fit

Transcription

1 10.2: The Chi Square Test for Goodness of Fit We can perform a hypothesis test to determine whether the distribution of a single categorical variable is following a proposed distribution. We call this type of hypothesis test a goodness of fit test. For testing goodness of fit, we consider a single population with one categorical variable having more than two categories (for two categories, use Chapter 8 methods). We want to know if the population distribution of the categories is as we propose, or if it is somehow different (we could propose that all categories occur with equal frequency, or maybe we have particular proportions we expect for each category). 1

2 Conditions: 1. The sample must be collected randomly and the observations within the sample must be independent. 2. The sample size is large (the expected count in each cell is at least 5). Recall the four step process: Step 1: Hypothesize State your hypotheses about the categorical variable. In general, this will look like: H 0 : The population distribution is the same as the proposed distribution H a : The distributions are different 2

3 Step 2: Prepare State a significance level. Choose an appropriate test statistic ( test statistic) State and verify all conditions are met. State any assumptions that must be made. Step 3: Compute and compare Compute the observed value of the test statistic. Find the p value (measure of surprise) using a distribution with degrees of freedom df = number of categories 1 NOTE: We will use Minitab to find the p value for any chi square test. Step 4: Interpret Do you have sufficient evidence to reject the null hypothesis (pvalue α)? Interpret results in context of the data. 3

4 Ex 1: We wish to determine whether child births occur in unequal proportions depending on the day of the week. We have a random sample of 773 births as listed below. Perform a goodness of fit analysis to test the hypothesis that births occur in different proportions each day of the week at the = 0.5 significance level. Step 1: Hypothesize Step 2: Prepare 4

5 Step 3: Compute and compare (We have already computed the chi square test statistic by hand. Now we will use Minitab to confirm this number and find the p value.) Type your observed counts in column C1. Select Stat, Tables, Chi Square Goodness of Fit Test (One Variable). Select Observed counts: and type C1 into the box next to the bullet. Then select Equal proportions. Click OK. 5

6 Chi Square Goodness of Fit Test for Observed Counts in Variable: C1 Test Contribution Category Observed Proportion Expected to Chi Sq N DF Chi Sq P Value Step 4: Interpret 6

7 Ex 2: The accuracy of a census report on a city in southern California was questioned by some government officials. The census claims that ethnicities occurred in the following proportions: African American 10%, Asian 3%, Anglo 38%, Latino 41%, Native American 6%, Other 2%. A random sample of 1215 people living in the city was used to check the report with the following results. Test the claim that the census is not accurate at the 0.05 significance level. Step 1: Hypothesize 7

8 Step 2: Prepare Step 3: Compute and compare Compute the observed value of the chi square statistic. 8

9 Type your observed counts in column C1 and the expected counts in C2. Select Stat, Tables, Chi Square Goodness of Fit Test (One Variable). Select Observed counts: and type C1 into the box next to the bullet. Then select Proportions specified by historical counts and choose C2. Click OK. Chi Square Goodness of Fit Test for Observed Counts in Variable: C1 Historical Test Contribution Category Observed Counts Proportion Expected to Chi Sq N DF Chi Sq P Value Step 4: Interpret 9