Stat 776 HW07
1. p261 5.1 (c) [5th edition: p260 5.1 (c)] by p-value and by rejection region
I by p-value
H0:µ=µ7
11 ¶Ha:µ6=µ7
11 ¶
Test Statistic: T2= (X−µ0)0(S
n)−1(X−µ0), µ0=µ7
11 ¶
p-value: P(F(2, n −2) >n−2
(n−1)2 T2
ob
T2= (−1,−1)[1
4µ8−3.3333
−3.3333 2 ¶]−1µ−1
−1¶= 13.64
p-value: P(F(2, n −2) >n−2
(n−1)2 T2
ob =P(F(2,2) >4.55) >0.10
(F0.10(2,2) = 9.00)
Fail to reject H0.
II by rejection region
H0:µ=µ7
11 ¶Ha:µ6=µ7
11 ¶
Test Statistic: T2= (X−µ0)0(S
n)−1(X−µ0), µ0=µ7
11 ¶
Reject H0if T2>(n−1)p
n−2Fα(p, n −p) = 3 ×19 = 57
T2= 13.64
Fail to reject H0.
2. p338 6.5 (a) [5th edition: p333 6.5 (a)] using rejection region
Let Cbe µ1−1 0
1 0 −1¶
H0:Cµ = 0 Ha:C µ 6= 0
Test Statistic: T2= (CX )0(C SC0
n)−1(CX )
Rejection region: T2>(n−1)q
n−qFα(q, n −q) = (40−1)×2
40−2F0.05(2,38) = 6.63
T2
ob = 90.5
Reject H0
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