Consider a random sample X,..., Xn from a Bernoulli distribution with pdf g(xp) = (1-P)-p, 0...

 

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Consider a random sample X₁,..., Xn from a Bernoulli distribution with pdf g(xp) = (1-P)¹-p², 0 < p < 1, z € (0, 1). Assume that we have collected 7 observations = (1,0,0,1, 1, 1,1). We would like to test Ho: Pop versus H₁: po #p. (a) [10 points] Test the hypotheses with p = 0.4 at a = 0.1 using the test statistic T(X) = X based on its exact distribution. (b) [10 points] Define the likelihood ratio test statistic A(z) and use it to conduct an asymptotic level-a test of Ho: Po = 0.4 versus H₁: Po 0.4 at a=0.1. Do you reject Ho? (c) [10 points] Assume that the tests from (a) and (b) led to different conclusions. Which one would you prefer? If you prefer one test, why and/or when would you ever use the other one? (d) [10 points] Use the asymptotic normality of maximum likelihood estimators and the plug- in information number ip to construct an asymptotic 1 - a confidence interval for po. Hint: You may use that PML = X and ip = +without deriving it yourself. (e) [5 points] Compute the interval from (d) for n = 70, * = 5/7 and a = 0.1. Interpret the computed interval and comment on whether the interval can be shortened. Hint: In case you were unsuccessful in deriving the interval in (d), you may use the incorrect 2¹/2 interval C(X) = [X - 2012, X+ Activate Wind Consider a random sample X₁,..., Xn from a Bernoulli distribution with pdf g(xp) = (1-P)¹-p², 0 < p < 1, z € (0, 1). Assume that we have collected 7 observations = (1,0,0,1, 1, 1,1). We would like to test Ho: Pop versus H₁: po #p. (a) [10 points] Test the hypotheses with p = 0.4 at a = 0.1 using the test statistic T(X) = X based on its exact distribution. (b) [10 points] Define the likelihood ratio test statistic A(z) and use it to conduct an asymptotic level-a test of Ho: Po = 0.4 versus H₁: Po 0.4 at a=0.1. Do you reject Ho? (c) [10 points] Assume that the tests from (a) and (b) led to different conclusions. Which one would you prefer? If you prefer one test, why and/or when would you ever use the other one? (d) [10 points] Use the asymptotic normality of maximum likelihood estimators and the plug- in information number ip to construct an asymptotic 1 - a confidence interval for po. Hint: You may use that PML = X and ip = +without deriving it yourself. (e) [5 points] Compute the interval from (d) for n = 70, * = 5/7 and a = 0.1. Interpret the computed interval and comment on whether the interval can be shortened. Hint: In case you were unsuccessful in deriving the interval in (d), you may use the incorrect 2¹/2 interval C(X) = [X - 2012, X+ Activate Wind

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Que A To test the hypotheses H0 tildep p0 versus H1 tildep eq p0 using the given observations x 1 0 0 1 1 1 1 we can use the test statistic TX overlineX where overlineX is the sample mean Given that t... View the full answer