Is this die fair? The chi-squared goodness-of-fit test

Question

You roll a die 60 times and observe face counts [8, 10, 9, 12, 11, 10]. Test whether the die is fair. Give the statistic, the degrees of freedom, and the conclusion.

Accepted Answer

Statistic. For k categories with observed counts Oi and expected counts Ei under the null, \chi^2 = \sum{i=1}^{k} \frac{(Oi - Ei)^2}{Ei}. Under H0 this is approximately \chi^2 with 	ext{df} = k - 1 - p, where p is the number of parameters estimated from the data. Here the null is fully specified (a fair die), so p = 0 and 	ext{df} = k - 1 = 5. Compute. A fair die gives Ei = 60/6 = 10 for every face. The squared deviations: \frac{(8-10)^2}{10} + \frac{(10-10)^2}{10} + \frac{(9-10)^2}{10} + \frac{(12-10)^2}{10} + \frac{(11-10)^2}{10} + \frac{(10-10)^2}{10} = \frac{4 + 0 + 1 + 4 + 1 + 0}{10} =…

Is this die fair? The chi-squared goodness-of-fit test

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