Question

INFERENCE ON CATEGORICAL DATA

1. For the given table of observed frequencies:

	1	2	3
A	15	10	12
B	3	11	11
C	9	14	12

1. Compute the value of the chi-square statistics.

2. If appropriate, perform a test of independence, using the α=0.05 level of significance.

Answer 1

A)

The Chi-Squared statistic is computed as follows:

B)

Chi-Square Independence test

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H0​: The two variables - ABC and 123 are independent Ha​: The two variables - ABC and 123 are dependent This corresponds to a Chi-Square test of independence. (2) Degrees of Freedom The number of degrees of freedom is df = (3 - 1) * (3 - 1) = 4 (3) Critical value and Rejection Region Based on the information provided, the significance level is α=0.05, the number of degrees of freedom is df = (3 - 1) * (3 - 1) = 4, so the critical value is 9.4877. Then the rejection region for this test becomes R={χ2:χ2>9.4877}. (4)Test Statistics The Chi-Squared statistic is computed as follows: (5)P-value The corresponding p-value for the test is p=Pr(χ2​>6.4808)=0.166 (6)The decision about the null hypothesis Since it is observed that χ2=6.4808<χ2_c​rit=9.4877, it is then concluded that the null hypothesis is NOT rejected. (7)Conclusion It is concluded that the null hypothesis Ho is NOT rejected. Therefore, there is NOT enough evidence to claim that the two variables - ABC and 123 are dependent, at the 0.05 significance level. Conditions: a. The sampling method is simple random sampling. b. The data in the cells should be counts/frequencies c. The levels (or categories) of the variables are mutually exclusive.

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