Question

Test the following hypotheses by using the χ ² goodness of fit test.

H 0:	p A = 0.2, p B = 0.4, and p C = 0.4
Ha:	The population proportions are not p A = 0.2 , p B = 0.4 , and p C = 0.4

A sample of size 200 yielded 60 in category A, 120 in category B, and 20 in category C. Use  = .01 and test to see whether the proportions are as stated in H₀. Use Table 12.4.

a. Use the p-value approach.

χ ² =  (to 2 decimals)

The p-value is Selectless than .005between .005 and .01between .01 and .025between .025 and .05between .05 and .10greater than .10Item 2

Conclusion:
SelectConclude the proportions differ from 0.2, 0.4, and 0.4.Cannot conclude that the proportions differ from 0.2, 0.4, and 0.4.Item 3

b. Repeat the test using the critical value approach.

χ ² _.01 =  (to 3 decimals)

Answer 1

The (one-way or goodness of fit) chi square contingency table would be as below.

	A	B	C
Observed Expected	60 200*0.2=40	120 200*0.4=80	20 200*0.4=80

(a) The chi square would be or . For a single row, the degree of freedom would be , for k being the number of categories. The p-value in this case is less than 0.005, which can be verified by any online calculator. What is not be noted is that it is less than 0.01, the alpha level.

Since the p value is less than alpha, we reject the null, and conclude that the proportions differ from 0.2, 0.4 and 0.4.

(b) The critical value would be as , which can be found by any online calculator or by simple R-command : "qchisq(1-0.01,2)".

Since calculated chi square is greater than the critical one, ie , we reject the null and conclude that the observed proportions were different from the expected proportion (0.2,0.4,0.4).