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: $30,29,46,39,29,27$. Use a 0.025 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? Click here to view the chi-square distribution table The test statistic is $\square$ (Round to three decimal places as needed.) The critical value is $\square$. (Round to three decimal places as needed) State the conclusion $H_{0}$. There sufficient evidence to support the claim that the outcomes are not equally likely. The outcomes to be equally likely, so the loaded die to behave differently from a fair die w an example Get more help - Time Remaining: 00:44:01 Next Search
Solution
Step 1 ：Define the null hypothesis as the outcomes are equally likely, which means each outcome (1,2,3,4,5,6) has an expected frequency of 200/6 = 33.33. The alternative hypothesis is that the outcomes are not equally likely.
Step 2 ：Calculate the test statistic and compare it with the critical value from the chi-square distribution table with 5 degrees of freedom (6 outcomes - 1) at a significance level of 0.025.
Step 3 ：Given observed frequencies are [30, 29, 46, 39, 29, 27] and expected frequencies are [33.333333333333336, 33.333333333333336, 33.333333333333336, 33.333333333333336, 33.333333333333336, 33.333333333333336].
Step 4 ：The critical value is 12.832501994030027.
Step 5 ：Compare the test statistic with the critical value. If the test statistic is greater than the critical value, reject the null hypothesis and conclude that the outcomes are not equally likely.
Step 6 ：The test statistic is \( \boxed{5.400} \) and the critical value is \( \boxed{12.833} \).
Step 7 ：There is not sufficient evidence to support the claim that the outcomes are not equally likely. The loaded die does not behave differently from a fair die.
