Consider an i.i.d. sample X1,…,Xn∼Poiss(λ) for λ>0 .

Question

Starting from the Central Limit Theorem, find a confidence interval I=[A,B] with asymptotic level 1−α that is centered about Xn using the plug-in method.

now consider the following hypothesis with a fixed number λ0>0 :

H0:λ=λ0vsH1:λ≠λ0.

Define a test for the above hypotheses with asymptotic level α , and rewrite it in the form

ψ=1{λ0∉J},

for some interval J=[C,D] .

( Write barX_n for Xn . If applicable, type abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α quantile of a standard normal variable. )

C=?
D=?