1. I have a 6 sided die which I claim is fair, i.e., that each of the 6 sides is equally likely to land facing up when rolled. You roll the die 11 times before obtaining a '6'. Should you conclude the die is not fair? (Hint: let X denote the roll of the die on which the first '6' appears; calculate P(X ≥ 11), this is your p-value).
2. One of my prized possessions is my 'simplified quantum coin', described by P(X = 1|p) = p, P(X = 0|p) = 1 - p, where X is the number of heads in a toss of the coin, and p is a random variable with P(p = 1/4) = 1/4 and P(p = 3/4) = 3/4.
(a) What is the probability of a 'head' in a single toss of this coin?
(b) If the coin is tossed and a 'head' results, what is the probability that p is 3/4?
7. Consider a random sample X1, ... , Xn" from a Uniform distribution on (0, θ). Derive a likelihood ratio test of H0 : θ = θ0 versus HA : θ ≠ θ0. Show that the rejection region is of the form X(n) ≤ cθ0 or X(n) > θ0 and find c for level α = 0.05.