4. Let Xl" ' " X; be independently distributed, each with probability p or q as N(t aJ) or Na, af). (i) If p is unknown, determine the UMP unbiased test of H : = 0 against K : ~> O. (ii) Determine the most powerful test of H against the alternative ~l when it is known that p = t, and show that a UMP unbiased test does not exist in this case. (iii) Let ak (k = 0, ... , n) be the conditional level of the unconditional most powerful test of part (ii) given that k of the X's came from Na, aJ) and n - k from N(t af). Investigate the possible values a o, ai' .. . , an"