Question

Conduct the hypothesis test and provide the test statistic and the critical​ value, and state the conclusion. A person drilled a hole in a die and filled it with a lead​ weight, then proceeded to roll it 200 times. Here are the observed frequencies for the outcomes of​ 1, 2,​ 3, 4,​ 5, and​ 6, respectively: 27​, 28​, 45​, 41​, 27​, 32. Use a 0.10 significance level to test the claim that the outcomes are not equally likely. Does it appear that the loaded die behaves differently than a fair​ die?

Answer 1

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H0: p1 =0.166666666666667, p2 =0.166666666666667, p3​=0.166666666666667, p4​=0.166666666666667, p5 =0.166666666666667, p6=0.166666666666667 Ha​: Some of the population proportions differ from the values stated in the null hypothesis This corresponds to a Chi-Square test for Goodness of Fit. (2) Degrees of Freedom The number of degrees of freedom is df=n-1=6-1=5 (3) Test Statistics The Chi-Squared statistic is computed as follows: (4)Critical Value and Rejection Region Based on the information provided, the significance level is α=0.1, the number of degrees of freedom is df=n-1=6-1=5, so the critical value becomes 9.2364. Then the rejection region for this test is R={χ2:χ2>9.2364}. (5)P-value The P-value is the probability that a chi-square statistic having 5 degrees of freedom is more extreme than 9.2364. The p-value is p=Pr(χ2>9.16)=0.1028 (6) Decision about the null hypothesis Since it is observed that χ2=9.16≤χc2​=9.2364, 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 some of the population proportions differ from those stated in the null hypothesis, at the α=0.1 significance level. Conditions a. The sampling method is simple random sampling. b. The variable under study is categorical. c. The expected value of the number of sample observations in each level of the variable is at least 5.

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