Question

Are the four predominant ethnicities in a town equally represented? A random sample of 13,442 town inhabitants is taken and are asked to identify themselves according to one of the following four ethnicities: 1 = Caucasian, 2 = African-American, 3 = Mexican American, 4 = Taiwanese-American. Does this sample provide evidence that the four races are equally represented in the town? A partial Minitab output for this question is below. Chi-Square Goodness-of-Fit Test for Categorical Variable: Ethnicity Test Category observed Proportion Expected 3476 0.25 3360.5 2 3234 10.25 3360.5 3 3080 0.25 3360.5 4 3652 0.25 3360.5 Contribution to Chi-sa 31.9697 4.7619 23.4133 25.2856 N 13442 N* DF 0 3 Chi-sq P-value 57.4304 0.000 Explain, in the context of this question, why this is a chi-square goodness-of-fit test. 1 i В. I = III do C

Answer 1

a) As we are testing here whether the sample provides enough evidence that the four races are equally likely represented in the town, therefore the null and the alternative hypothesis here are given as:

H0: The four races are equally likely represented in the town

Ha: The four races are not equally likely represented in the town

b) There are 3 conditions that need to be satisfied for the goodness of fit test and all of them are satisfied here. These are given as:

• There is simple random sampling done
• The expected Frequencies for each category must be at least 5, they all are 3360.5 which are greater than 5
• The variable under study is categorical ( here there are 4 categories )

c) The chi square test statistic here is given to be 57.4304 and the p-value as 0.000 ( as seen from the given Minitab output )

d) As the p-value here is 0.000 which is very very low, therefore the test is significant here and we can reject the null hypothesis here. Therefore we have sufficient evidence here that The four races are not equally likely represented in the town