Problem 1 A gambler observing a game in which a single die is tossed repeatedly gets the impression that 6 comes up about 18% of the time, 5 about 14% of the time, while the other 4 numbers are equally likely to occur (i.e., with probability 17%). Upon being asked to play, the gambler asks that he first be allowed to test his hypothesis by tossing the die n times. 1. What test statistic should he use if the only alternative he considers is that the die is fair? 2. Show that if n = 2 the most powerful level 0.0196 test rejects the null hypothesis if, and only if, two 5's are obtained. 3. Using the fact that if (N1, ... ,Nk) ~ M(n, 01, ... , Ok), the multinom- inal distribution, then when n is large enough, aNi + ... + axNx has approximately a N(np, no2) distribution with j = El= a;b; and o2 = !-10;(ai u)?, find an approximation to the critical value of the most powerful test for this problem at level a. Problem 1 A gambler observing a game in which a single die is tossed repeatedly gets the impression that 6 comes up about 18% of the time, 5 about 14% of the time, while the other 4 numbers are equally likely to occur (i.e., with probability 17%). Upon being asked to play, the gambler asks that he first be allowed to test his hypothesis by tossing the die n times. 1. What test statistic should he use if the only alternative he considers is that the die is fair? 2. Show that if n = 2 the most powerful level 0.0196 test rejects the null hypothesis if, and only if, two 5's are obtained. 3. Using the fact that if (N1, ... ,Nk) ~ M(n, 01, ... , Ok), the multinom- inal distribution, then when n is large enough, aNi + ... + axNx has approximately a N(np, no2) distribution with j = El= a;b; and o2 = !-10;(ai u)?, find an approximation to the critical value of the most powerful test for this problem at level a