478 Optimal Tests of HypothesesEXERCISES8.1.1.In Example 8.1.2 of this section, let the simple hypotheses readH 0 :θ=
θ′=0andH 1 :θ=θ′′=−1. Show that the best test ofH 0 againstH 1 may be
carried out by use of the statisticX,andthatifn=25andα=0.05, the power of
the test is 0.9996 whenH 1 is true.
8.1.2.Let the random variableXhave the pdff(x;θ)=(1/θ)e−x/θ, 0 <x<∞,
zero elsewhere. Consider the simple hypothesisH 0 :θ=θ′= 2 and the alternative
hypothesisH 1 :θ=θ′′=4. LetX 1 ,X 2 denote a random sample of size 2 from this
distribution. Show that the best test ofH 0 againstH 1 maybecarriedoutbyuse
of the statisticX 1 +X 2.
8.1.3. Repeat Exercise 8.1.2 whenH 1 :θ=θ′′= 6. Generalize this for every
θ′′>2.
8.1.4.LetX 1 ,X 2 ,...,X 10 be a random sample of size 10 from a normal distribution
N(0,σ^2 ). Find a best critical region of sizeα=0.05 for testingH 0 :σ^2 = 1 against
H 1 :σ^2 = 2. Is this a best critical region of size 0.05 for testingH 0 :σ^2 = 1 against
H 1 :σ^2 = 4? AgainstH 1 :σ^2 =σ^21 >1?
8.1.5.IfX 1 ,X 2 ,...,Xnis a random sample from a distribution having pdf of the
formf(x;θ)=θxθ−^1 , 0 <x<1, zero elsewhere, show that a best critical region
for testingH 0 :θ= 1 againstH 1 :θ=2isC={(x 1 ,x 2 ,...,xn):c≤
∏n
i=1xi}.8.1.6.LetX 1 ,X 2 ,...,X 10 be a random sample from a distribution that isN(θ 1 ,θ 2 ).
Find a best test of the simple hypothesisH 0 :θ 1 =θ 1 ′=0,θ 2 =θ 2 ′= 1 against the
alternative simple hypothesisH 1 :θ 1 =θ 1 ′′=1,θ 2 =θ 2 ′′=4.
8.1.7. LetX 1 ,X 2 ,...,Xndenote a random sample from a normal distribution
N(θ,100). Show thatC={(x 1 ,x 2 ,...,xn):c≤x=
∑n
1 xi/n}is a best critical
region for testingH 0 :θ= 75 againstH 1 :θ= 78. Findnandcso that
PH 0 [(X 1 ,X 2 ,...,Xn)∈C]=PH 0 (X≥c)=0. 05and
PH 1 [(X 1 ,X 2 ,...,Xn)∈C]=PH 1 (X≥c)=0. 90 ,approximately.
8.1.8.IfX 1 ,X 2 ,...,Xnis a random sample from a beta distribution with param-
etersα=β=θ>0, find a best critical region for testingH 0 :θ= 1 against
H 1 :θ=2.
8.1.9. LetX 1 ,X 2 ,...,Xnbe iid with pmff(x;p)=px(1−p)^1 −x,x=0,1, zero
elsewhere. Show thatC={(x 1 ,...,xn):
∑n
1 xi≤c}is a best critical region for
testingH 0 :p=^12 againstH 1 :p=^13. Use the Central Limit Theorem to findn
andcso that approximatelyPH 0 (
∑n
1 Xi≤c)=0.10 andPH 1 (∑n
1 Xi≤c)=0.80.