The gambler rolled a die 120 times and observed frequencies that differed from the expected values of 20 for each outcome if the die was fair. A chi-square test was conducted to compare the observed and expected values. The results showed that the die was not fair at the 0.05 significance level but could not be rejected as unfair at the more stringent 0.01 level, as the observed chi-square value fell between the critical values for those levels.

1. A gambler suspects that a pair of dice he has been playing with are loaded. He rolls one of the
dice 120 times and observes the follow frequencies: one, 30; two, 17; three, 27; four, 14; five,
13; six, 19. Can he reject the null hypothesis that the die is fair at the .05 level? What about at
the .01 level?
Solution
Note: All expected values are (1/6)*120 = 20. Chi-square = sum from i = 1 to n of
(Oi - Ei)^2 / Ei = (30-20)^2 / 20 + (17-20)^2 / 20 + (27 - 20)^2 / 20 + (14 - 20)^2 / 20 + (13 -
20)^2 / 20 + (19 - 20)^2 /20 = 12.2 Critical chi-squares for df = n-1 = 5 for .05 and .01 are,
respectively: 11.07 and 15.09. Since the observed chi-square is greater than 11.07 but less than
15.09, he can reject the null hypothesis at the .05 level but not at the .01 level.