Testing Whether Random Sample is From a Discrete Uniform Population Distribution. Let \(X_{i}, i=1, \ldots, n\) be random sample from some discrete integer-valued population distribution. Let the range of the random variables be \(1,2, \ldots, m\).

(a) Describe how you would use the \(\chi^{2}\) goodness of fit size \(\alpha\) test to assess the hypothesis that the population distribution was a discrete uniform distribution with support \(1,2, \ldots, m\).

(b) The Nevada Gaming Commission has been called in to investigate whether the infamous Dewey, Cheatum, \& Howe Casino is using fair dice in its crap games. One of the alleged crooked dice is tossed 240 times, and the following outcomes were recorded:

Does this die appear to be fair? What size test will you use? Why?

X 1 2 3 4 5 6 Frequency 33 44 47 39 31 46