If the dieis fair, then all six outcomes (1, 2, 3, 4, 5, 6) should be equally likely, therefore the probability of getting any specific value should be 1/6 (or 0.17 rounded to 2 decimal places).

Let X represents the outcome of a die roll and P(X=1) represent the probability that a single die roll would produce an outcome of 1. Then P(X=1) = P(X=2) = P(X=3) = P(X=4) = P(X=5) = P(X=6) = 1/6

Suppose you roll the die 36 times and recorded the outcome of each roll:1 5 2 3 4 3 1 3 6 1 5 6 2 1 1 3 3 5 1 5 2 3 6 5 5 6 3 4 5 1 3 2 2 1 6 3

A chi-square test can then be used to assess whether this observed distribution of die rolls (categorical data) significantly deviates from the expected distribution of die rolls if the die is fair.

For this test, the null hypothesis, H₀, is that the die is fair and the alternate hypothesis, H₁, is that the die is NOT fair.

The chi-square test statistic is denoted as ² (looks a little like a big X, but it's not actually and X) is calculated using the equation² = [(O-E)² / E].

a) What is the chi-square test statistic for assessing whether the 36 observed die rolls (shown above) suggest the die is fair? Enter the answer below in decimal form, round to 2 decimal places.

b)The degrees of freedom for the chi-square goodness of fit test is the number of possible outcomes minus 1. What is the df for the chi-square test statistic in the last question?