I’m working on a probability question and need guidance to help me learn.
Let X1; : : : ;X120 be a collection of i.i.d. random variables and let Y1; : : : ; Y120 be the associated
order statistics.
(a) [4 points] Find a value of j such that the interval [Y30..j ; Y30+j ] is an approximate 95%
condence interval for the 1
4 quartile of the distribution.
Problem 6. 12 points
Suppose X1; : : : ;X60 are i.i.d. b(1; p) random variables where p is unknown. You are given the
null hypothesis H0 : p = 1=4.
(a) [3 points] Suppose you want to test H0 against the simple alternative hypothesis H1 : p =
1=2. Use the likelihood ratio test to nd a critical region C of (approximate) size :005 (i.e.
the probability of a type 1 error is :005).
(b) [4 points] Show that the likelihood ratio test gives you get the same critical region C from
part (a) if the alternative hypothesis is H1 : p = p1 where p1 is any number bigger than 1
4 .
(c) [3 points] What is the type 2 error associated with the region C when the alternative
hypothesis is p = 1
3?
(d) [2 points] Brie
y explain why any other critical region D with size :005 must make a
larger type 2 error than C when the alternative hypothesis is p = 1
0 comments