Question

Let X 1, X 2, X 3, X 4 be a random sample of size n=4 from a Poisson distribution with mean . We wish to test Ho: I = 3 vs. H1: \<3. a) Find the best rejection region with the significance level a closest to 0.05. Hint 1: Since H1: X< 3, Reject Ho if X 1+X 2 +X 3 +X 4<= 0 Hint 2: X 1+X 2 +X 3 + X 4 ~ Poisson (4) Hint 3: Want C such that 0.05 a = P (Reject Ho Ho is true) =P (X 1+X 2 +X 3+X 4<=c|\ =3) Hint 4: Poisson distribution is discrete, integer-valued. b) Find the power of the test from part (a) (i) at = 2, (ii) at = 1.5 Hint: Power (N) =P (Reject Ho | Ho is NOT true) = P(X 1+X 2+X 3 +X4<=0|N). c) Suppose xl = 1, x2 = 3, x3 = 1, x4 = 2. Find the p-value of the test Hint: p-value =P (as extreme or more extreme He true)

Answer 1

(a) P(X1 + X2 + X3 + X4 <cH) < 0.05 Since under Ho, T = X1 + X2 + X3 + X4 ~ Poisson(12) then c= 6 Since PT < 6 H) = 0.0458 (U se R code: ppois(6, 12)) and P(T<7|H) = 0.0895 (U se R code: ppois(7, 12)) (6) Power (2) = P(T < 6T ~ Poisson(8)) = 0.3134 (U se R code: ppois(6,8)) Poisson (6)) = 0.6063 Power(1.5) = PT < 6T (U se R code: ppois(6, 6)) (c)21 + x2 +13 +24=1+3+1+2 = 7 p - value = P(T<7T ~ Poisson(12)) = 0.0895.