Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 6H (Statistics)

Writing $p_j$ for the probability of face $j$, the hypotheses areagainst

For observed counts $O_j$ and null expected counts $E_j=n{p}_j$, the Pearson chi-squared goodness-of-fit test usesWhen the null cell probabilities are specified and the expected counts grow, its limiting null distribution isHere $k=6$, so the limit is ${\chi }_5^2$.

Under fairness every expected count is $90/6=15$. Therefore

The asymptotic p-value is the upper-tail probabilityNumerically this is approximately $0.392$, so the data provide little evidence against fairness.