In a traditional Yahtzee game, five dice are tossed and the player tries to complete certain categories some of which are similar to poker like three of a kind, full house, or a straight.  If the player gets all five dice to be all the same number, it is called a Yahtzee.  There are hand held electronic versions of this game.  In this game, dice are not actually tossed - the built in computer simulates this.  This might make one wonder if these simulated dice are really random.  To answer this question, a hand held electronic Yahtzee was used to simulate drawing five dice 100 times.  The results can be found here.  We are going to conduct two types of chi-square tests on this data set.  The first will be to see if there is a relationship between the simulated die and the number shown on the die.  The second will be to see if each simulated die gives an even distribution of outcomes in the long run.

1. To answer the question of whether or not there is a relationship between the simulated dice and the number shown, we will do a chi-square independence test.  After putting the Yahtzee data  into Minitab, answer the following questions.
1. Considering all the dice, what percent of the total number of outcomes are ones?  What percent are twos?  What percent are threes?  What percent are fours?  What percent are fives?  What percent are sixes?  Based on these percentages, do you think there may be a relationship between the dice and the number shown?
2. Complete a chi-square test to determine if there is a relationship between each die and the number shown.  Use (Stat > Tables > Chi-square Test.)  Make sure you write out your hypotheses, give the chi-square statistic, the P-value, and a conclusion.

2. Another type of chi-square test is a goodness-of-fit test.  Instead of testing the entire two-way table for independence as done in question 1, this test can determine whether each die is gives an even distribution or not.  Minitab does not do this test automatically.  We are going to have to do some work to calculate the chi-square test statistic and P-value.  Remember that the formula used to calculate the chi-square test statistic is as follows.

1. We want to perform a chi-square goodness of fit test on each column to determine if each is not distributed evenly.  Write down the null and alternative hypotheses for this kind of test.
2. The expected frequencies for each outcome is 100/6 or approximately 16.67.  Use this to calculate the chi-square test statistic for Die_A.  To do this, choose Calc > Calculator.  In Store result in variable:, enter the name of an open column.  In Expression:, enter SUM((Die_A-16.67)**2/16.67). Click OK.  Repeat this for the other four dice.  Report your five chi-square test statistics.
3. We now need to calculate the P-values for each of our tests.  To do this select Calc > Probability Distributions > Chi-Square.  In the window that pops up click on Cumulative Probability, leave 0.0 as the Noncentrality parameter, put in 4 for your Degrees of freedom, click on Input constant, input your chi-square statistic for Die_A from part (b), and click OK.  The value that Minitab gives you is one minus the P-value, so subtract your output from one to get the P-value.  Repeat this for the other four dice. 
   
4. What are your conclusions for each of your five goodness-of-fit tests?
5. The mean of a chi-square distribution is equal to its degrees of freedom.  This means that if the null hypothesis is correct, we complete many different chi-square tests, we would expect the mean of our test statistics to be approximately the same as the degrees of freedom.  What is the mean of your chi-square test statistics from part (c)?  How does this compare with the number of degrees of freedom?
   

3. Based on the two different chi-square tests run on the Yahtzee data set, do you think the numbers shown on each die are distributed evenly?  Explain.