Question

Consider the next 1000 80% CIs for μ that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of μ?

What is the probability that between 790 and 810 of these intervals contain the corresponding value of μ? [Hint: Let Y = the number among the 1000 intervals that contain μ. What kind of random variable is Y?] (Round your answer to four decimal places.)

Answer 1

Solution:- 80% CI expects to capture u 80% of time

(a) E(X) = n*p = 1000*0.8 = 800

(b) P(790 <= x <= 810) = P(789.5 < X < 810.5)

= P(789.5 < X < 810.5)
= P((789.5-800)/sqrt(1000*0.8*0.2) < Z < (810.5-800)/sqrt(1000*0.8*0.2))
= P(-0.83 < Z < 0.83)
= 0.5934