Question

You must decide which of two discrete distributions a random variable X has. We will call the distributions p₀ and p₁. Here are the probabilities they assign to the values x of X:

x	0	1	2	3	4	5	6
p0	0.1	0.1	0.2	0.3	0.1	0.1	0.1
p1	0.1	0.3	0.2	0.1	0.1	0.1	0.1

You have a single observation on X and wish to test

H₀: p₀ is correct

H_a: p₁ is correct

One possible decision procedure is to reject H₀ only if X ? 1.

a) Find the probability of a Type I error, that is, the probability that you reject H₀ when p₀ is the correct distribution.

b) Find the probability of a Type II error.

Suppose that instead of a single observation X, you obtained two observations and use the decision rule to reject when ¯x ? 1.

c) Under this scenario, would you expect the probabilities of a Type I and Type II errors to increase, decrease, or stay at the same values of Exercise 6.120? Explain your answer.

d) Verify your answer to part (c) by computing the probabilities of a Type I and Type II error.

Answer 1

a) probability of a Type I error = P( X = 1, under the null hypothesis ) = 0.1

b) Find the probability of a Type II error

The power of the test = 1 - = P( X = 1 , under the alternative hypothesis ) = 0.3

Therefore type II error = = 1 - 0.3 = 0.7

c) Since the size of critical region increases so Type I error increase and Type II error decrease

d) Now critical region is X <= 1

Therefore type I error = P(X = 0 or 1, under the null hypothesis ) = 0.1 + 0.1 = 0.2

and Power of the test is = 1 - = P(X = 0 or 1, under the alternative hypothesis ) = 0.1 + 0.3 = 0.4

Therefore type II error = 1 - 0.4 = 0.6