Question 1. Let X~B(n, p), for the test problem H_o: p=0.2↔H_a:p=0.5, let the rejection region be X≥6.

a. if you have a sample of 12 (namely n=12), find the α and β;

b. in order to reduce β to be less than 5%, how large n you have to have?

Question 2. Let x₁,..., x₁₆ be an independent sample from a normal distribution N(μ, 5). For the following test problem: Ho: μ = 6.5 ↔Ha: μ ≠6.5, let the rejection region be W = {|x^-6.5| ≥ c} (Note, in the lecture, I often call the rejection region as "the wired region", and it was in term of our best "friend" z, like {z > 1.6} for instance. Here the rejection region is in term of x^-, but z and x^- are related.). Find c to make the significant level, namely α, of this test be 0.05. If μ = 7.5 is correct, find the probability of the second type error β.

Question 3. A, B, and C are three countries that have Ebola epidemics. In average, three out of 100000 persons is infected in A, one out of 100000 is infected in B, and one out of 200000 is infected in C. Some blood samples are sent to a lab for testing. In order to avoid bias, the tester is told that these blood samples come from one of the above countries but is not told which one.
a. The tester tests the first two samples, they are negative. What is the probability that the third sample will be tested positive?
b. The tester tests the first two samples, they are positive. What is the probability that the third sample will be tested negative?

Question 4. Let x₁, x₂ are two independent sample from population N(0,1), find such that .

P((x₁+x₂)²/(x₁-x₂)²+(x₁+x₂)² > k) = 0.05.

(Hint: both x1+x2 and x1-x2 are normal: N(0, √2))

Question 5. You are lost on the Mythy Island in the summer, when tourists are two-thirds of the population. If you ask a tourist for direction the answer is correct with probability ¾; answers to repeated questions are independent even if the question is the same. If you ask a local for directions, the answer is always false.

a. You ask a passer-by whether Mythy City is East or West. The answer is East. What is the probability that it is correct?

b. You ask her again, and get the same reply. What is the probability that it is correct?

c. You ask her one more time, and the answer is East again. What is the probability that it is correct? (Hint: Note, this problem is related to the problem of "False Positive".)