PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
To choose c, we use the fact that
Hence, the usual z-test,
Z = ¯X − µ 0
σ/ √ n ∼ N(0, 1) under H 0.
reject H 0 if ẑ ≥ z(α)
is the Neyman-Pearson LR test in this case.
Remarks
(1) This result shows that the one-sided z test is also uniformly most powerful
for
H 0 : µ = µ 0 vs. H A : µ > µ 0
(2) This can be extended to the case of
In this case we take
H 0 : µ ≤ µ 0 vs. H A : µ > µ 0
α = max
H 0
P (rejecting |µ = µ 0 ).
(3) This construction fails when we consider two-sided alternatives
i.e., H 0 : µ = µ 0 vs. H A : µ ≠ µ 0
=⇒ no uniformly most powerful test exists for that case.
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