Let X 1 , X 2 ,...,X 8 be a random sample of size n = 8 from a Poisson distribution with mean μ. Reject the simple null hypothesis H0 : μ = 0.5 and accept H 1 : μ > 0.5 if the observed sum . (a) Show...

Let X₁, X₂,...,X₈
be a random sample of size n = 8 from a Poisson distribution with mean μ. Reject the simple null hypothesis H0 : μ = 0.5 and accept H₁
: μ > 0.5 if the observed sum
.

(a) Show that the significance level is 1-ppois(7,8*.5).

 (b) Use R to determine γ(0.75), γ(1), and γ(1.25).

Let Y have a binomial distribution with parameters n and p. We reject H₀
: p = 1/2 and accept H₁
: p > 1/2 if Y ≥ c. Find n and c to give a power function γ(p) which is such that γ(1/2) = 0.10 and γ(3/4) = 0.95, approximately.