Let X, X, be i.i.d. samples from Bernoulli(p). We want to find a uniformly most powerful...

 

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Let X₁, X, be i.i.d. samples from Bernoulli(p). We want to find a uniformly most powerful test for the null hypothesis Ho: p = po against the one-sided alternative hypothesis H₁: P= P1 > Po. (i) Compute the ratio L(po)/L(p₁) of joint likelihood functions, where p₁> po. Show that 11- Po L(po) L(p₁) sk Po(1-P₁) pi(1-po). Exzc: i=1 sk logk-nlog[(1-Po)/(1-P)] log[po(1-P₁)/pi(1-po)] (3) (4) (ii) Using the Neyman-Pearson lemma, conclude that a best critical region C for testing Ho: p = Po against H₁ : p = P₁ > Po is given by c={(x,x) | Ex=c}. C= (iii) Let n = 30. Use CLT and (ii) and obtain a best critical region C for testing Ho: p = 0.2 against H₁: p=0.5 at significance level a = 0.05. (5) Let X₁, X, be i.i.d. samples from Bernoulli(p). We want to find a uniformly most powerful test for the null hypothesis Ho: p = po against the one-sided alternative hypothesis H₁: P= P1 > Po. (i) Compute the ratio L(po)/L(p₁) of joint likelihood functions, where p₁> po. Show that 11- Po L(po) L(p₁) sk Po(1-P₁) pi(1-po). Exzc: i=1 sk logk-nlog[(1-Po)/(1-P)] log[po(1-P₁)/pi(1-po)] (3) (4) (ii) Using the Neyman-Pearson lemma, conclude that a best critical region C for testing Ho: p = Po against H₁ : p = P₁ > Po is given by c={(x,x) | Ex=c}. C= (iii) Let n = 30. Use CLT and (ii) and obtain a best critical region C for testing Ho: p = 0.2 against H₁: p=0.5 at significance level a = 0.05. (5)

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