a. For the alternative hypothesis, you normally would want to choose something that says not equal since you will need to be testing if the proportions are equal and this will be your null hypothesis.
Therefore for your alternative hypothesis it would need to be: Not all population proportions are equal to 1/6.
b-1. Since we have 6 different outcomes, we will use the chi square test to evaluate the hypothesis. The test statistic is computed by getting the summation of [tex] \frac{ (observed-expected)^{2} }{expected} [/tex] for all the 6 outcomes. In our case the expected proportion would be 1/6 or 0.1667 and the observed values would be the proportions from the 6 outcomes.
[tex] X^{2} = \frac{ (0.1860-0.1667)^{2} }{0.1667} +\frac{ (0.2016-0.1667)^{2} }{0.1667}+\frac{ (0.1977-0.1667)^{2} }{0.1667}[/tex]
[tex]+\frac{ (0.1318-0.1667)^{2} }{0.1667}+\frac{ (0.1860-0.1667)^{2} }{0.1667}+\frac{ (0.0969-0.1667)^{2} }{0.1667}=0.0541[/tex]
ANSWER: The test statistic is 0.0541
b-2. For this problem we just imagine the chi square distribution and the fact that our test statistic is so close to zero so the p-value must be so much greater than 0.10.
Based on other resources, there are choices to this problem so we just choose the option similar to the one we just described.
ANSWER: p-value > 0.10
c. Since the p-value we estimated above is greater than 0.10, we do not reject the null hypothesis. This means that there is NOT enough evidence to reject the fact that the die is a fair die.
ANSWER: No, we cannot conclude that the die is loaded.
