D Question 8 11 pts I have an unfair coin that shows heads only 25% of the time. We want to play a game, but you don't know whether I am using the unfair coin or a fair one. You suspect I am cheating. However, that's a strong statement and you want to test it, and accuse me of cheating only if you have enough evidence. So, you would want to test Ho : 0 = 5 vs. HA : 0 = To test this, suppose you are allowed to toss a coin as many times as needed until heads appears for the first time (and then you stop). So, you use the statistic X = number of tosses of the coin until heads appears as an outcome (for the first time). Note that if the coin is unfair you would expect to wait longer, i.e. toss the coin more times until heads appears for the first time. Which of these sets C is the most reasonable choice for the critical (i.e. rejection) region? Also, for the right choice of the critical region, how would you find the probability of a type I error a = probability of (incorrectly) rejecting Ho when it is actually true? Check all that apply (minor hint: check exactly two boxes). Hint: think about what values x (= number of tosses until head appears for the first time) would give you a good evidence that the die is loaded, and what values a would be evidence that the coin is more likely to be fair.11 pts D Question 7 A fabric manufacturer believes that the proportion p of orders arriving late from a certain supplier of raw material exceeds 20%. To test this claim, the manufacturer collects a sample and records X = the number of orders in the sample that had been late. You can assume the orders are independent of each other, in terms of tardiness. The manufacturer wants to bring this problem up and confront the supplier only if they are certain in the claim with significance level of 10%. Which of the following tests should the manufacturer perform? O Ho : p 0.1; critical region: X k for some whole number k. O Ho : P 0.2; critical region: X 0.1; critical region: X > k for some whole number k. O Ho : p 0.2; critical region: X _ k for some whole number k.D Question 6 11 pts A discrete random variable X has probability mass function f(z) = P( X = =) = e-. #, A , x = 0, 1, 2, 3, ... where A is unknown parameter. Using a collected sample (X1, 12, . . ., , In ) of size n, find AMLE , the maximum likelihood estimate of the parameter 1. O AMLE = O AMLE = 1 O AMLE = O AMLE = EN Ti O AMLE =