A researcher wanted to test the mean of the binomial distribution as H0 : λ = 2 vs H1 : λ = 1. He decided to take two samples and he will reject the null hypothesis if both of the samples are less than 2. a) Find the type 1 error of the test b) Find the type 2 error of the test I want you to use this example to answer the question: A researcher wanted to test the mean of the Poisson distribution as H0 : λ = 2 vs H1 : λ = 1. He decided to take two samples and he will reject the null hypothesis if both of the samples are less than 2.

A researcher wanted to test the mean of the binomial distribution as H0 : λ = 2 vs H1 : λ = 1. He decided to take two samples and he will reject the null hypothesis if both of the samples are less than 2.

a) Find the type 1 error of the test

b) Find the type 2 error of the test

I want you to use this example to answer the question:

A researcher wanted to test the mean of the Poisson distribution as H0 : λ = 2 vs H1 :
λ = 1. He  decided to take two samples and he will reject the null hypothesis if both of the samples
are less than 2.
(a) Find the type one error of the test.
Solution: The null hypothesis is rejected if both the samples are less than 2.
So RR = {Y1 < 2 ∩ Y2 < 2} where Yi ∼ P oi(λ)
Now Type-I error = α = P (Y1 < 2 ∩ Y2 < 2 | λ = 2)
= P (Y1 < 2) × P (Y2 < 2) = 0.406 × 0.406 = 0.1648
P (Y1 < 2|λ = 1) = ppois(1,2) = 0.406 (R-Code)
(b) Find the type-II error of the test. As Type-II error = 1 - Power, we will find power first.
Given the above rejection region
Power = = P (Y1 < 2 ∩ Y2 < 2 | λ = 1)
= P (Y1 < 2) × P (Y2 < 2) = 0.736 × 0.736 = 0.5417
P (Y1 < 2|λ = 1) = ppois(1,2) = 0.7357 (R-Code)
Hence Type-II error = 1 - Power = 1 - 0.5417 = 0.4583