Determine the power function of the test

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Reference no: EM132551149

Question (a) A random sample {X₁, X₂,....X_n} is drawn from the following probability distribution:

f(x; θ) = {2x/ (1-θ²) for θ ≤ x ≤ 1
0 otherwise.
i. Derive the maximum likelihood estimator of θ, and explain why this is a maximum.
ii. If n = 4, such that x₁ = 0.70, x₂ = 0.92, x₃ = 0.21 and x₄ = 0.34, determine the maximum likelihood estimate of O.
iii. As n → ∞, would you expect the maximum likelihood estimate of B to get closer to 6? Briefly explain your answer.

(b) A random sample of size n = 9 drawn from a normal distribution had a sample variance of s² = 72.76. Construct a 90% confidence interval for σ².

(c) Let X be a random variable such that X ~ Exp(λ). Suppose we wish to test the null hypothesis Ho : λ = 2 vs. H1 : λ > 2. Consider the test procedure which rejects Ho if X < 0.75.
i. Compute the probability of a Type I error, expressing your answer in terms of the exponential constant e.
ii. Determine the power function of the test, expressing your answer in terms of the exponential constant e.

(b) A random sample of size n = 9 drawn from a normal distribution had a sample variance of 62 = 72.76. Construct a 90% confidence interval for o2.

(c) Let X be a random variable such that X ". Exp(A). Suppose we wish to test the null hypothesis Ho : A = 2 vs. HI : A > 2. Consider the test procedure which rejects Ho if X < 0.75.

i. Compute the probability of a Type I error, expressing your answer in terms of the exponential constant e.

ii. Determine the power function of the test, expressing your answer in terms of the exponential constant e.

(e) In a game two fair dice are thrown. If the sum of the two dice is odd they are thrown once more. What is the probability that the dice in any final throw in the game show the same number?

Reference no: EM132551149

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