Following the same procedure the power of the test defined by C1 is as fol- wing for all e∈白1: Pr(r(X)|≥196:6=56)=0.8849 Pr(r(X)|≥196:6=58)=0.3520; Pr((X)≥1.96:6=60)=0.0500 Pr(r(X)川≥1.96;6=62)=0.3520; Pr(r(X)川≥1.96;6=64)=0.8849 Pr((x)|≥1.96;6=66)=0.9973 As we can see, the power of the test increase as we go further away from 8=60(Ho) and the power at 0=60 equals the probability of type I error. This prompts us to define the power function as follows Definition 2 P(0)=Pr(x E C1), 0E0 is called the power function of the test defined by the rejection region C1 Definition 3 a=matBee P(0) is defined to be the size (or the significance level)of the test In the case where Ho is simple, say 0=8o, then a=P(Bo) Definition 4 A test of Ho: 0 E 0o against H1: 0 E 01 as defined by some rejection region C1 is said to be uniformly most powerful(UMP)test of size a if (a) ma. F(⊙) (b)P(6)≥P(6) for all e∈O1, where P*(0)is the power function of any other test of size a As will be seen in the sequential, no UMP tests exists in most situations of interest in practice. The procedure adopted in such cases is to reduce the classFollowing the same procedure the power of the test defined by C1 is as fol￾lowing for all θ ∈ Θ1: Pr(|τ (X)| ≥ 1.96; θ = 56) = 0.8849; Pr(|τ (X)| ≥ 1.96; θ = 58) = 0.3520; Pr(|τ (X)| ≥ 1.96; θ = 60) = 0.0500; Pr(|τ (X)| ≥ 1.96; θ = 62) = 0.3520; Pr(|τ (X)| ≥ 1.96; θ = 64) = 0.8849; Pr(|τ (X)| ≥ 1.96; θ = 66) = 0.9973. As we can see, the power of the test increase as we go further away from θ = 60(H0) and the power at θ = 60 equals the probability of type I error. This prompts us to define the power function as follows. Definition 2: P(θ) = Pr(x ∈ C1), θ ∈ Θ is called the power function of the test defined by the rejection region C1. Definition 3: α = maxθ∈Θ0P(θ) is defined to be the size (or the significance level) of the test. In the case where H0 is simple, say θ = θ0, then α = P(θ0). Definition 4: A test of H0 : θ ∈ Θ0 against H1 : θ ∈ Θ1 as defined by some rejection region C1 is said to be uniformly most powerful (UMP) test of size α if (a) maxθ∈Θ0P(θ) = α; (b) P(θ) ≥ P ∗ (θ) for all θ ∈ Θ1, where P ∗ (θ) is the power function of any other test of size α. As will be seen in the sequential, no UMP tests exists in most situations of interest in practice. The procedure adopted in such cases is to reduce the class 8