Final answer:
To test the claim that the outcomes of rolling the loaded die are not equally likely, we can use a goodness-of-fit test. We compare the observed and expected frequencies and calculate the chi-square statistic. We then compare the test statistic to the critical value.
Step-by-step explanation:
To test the claim that the outcomes of rolling the loaded die are not equally likely, we can use a goodness-of-fit test. We will compare the observed frequencies with the expected frequencies for each outcome. The null hypothesis for this test is that the outcomes are equally likely, and the alternative hypothesis is that the outcomes are not equally likely.
To perform the goodness-of-fit test, we need to calculate the expected frequencies for each outcome. Since there are 6 possible outcomes and a total of 200 rolls, we would expect each outcome to occur approximately 200/6 = 33.333 times. Therefore, the expected frequencies for each outcome would be 33.333 * 1/6 = 5.556 for 1, 2, 3, 4, 5, and 6.
Now we can calculate the test statistic. The test statistic for the goodness-of-fit test is the chi-square statistic. It is calculated using the formula: chi-square = Σ ((observed frequency - expected frequency)^2 / expected frequency). We can substitute the observed frequencies provided in the question and the expected frequencies we calculated to find the chi-square statistic. Finally, we compare the chi-square statistic to the critical value of the chi-square distribution with 5 degrees of freedom at a significance level of 0.01.