practice_final_stat_solutions

practice_final_stat_solutions - SDS 321 Practice...

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SDS 321: Practice questions–Statistics 1. I am trying to estimate the average lifespan of elephants in zoos, using a Bayesian method. I assume that elephant lifespans are normally distributed with unknown mean μ and variance 9. Based on the lifespan of wild elephants, I put a Normal (25 , 25) prior on μ . I ask the zoo for data. They tell me their last 4 elephants lived to 18, 19.5, 17 and 15.5, respectively. What are the posterior mean and variance of μ ? mean: μ 0 σ 2 0 + i x i σ 2 1 σ 2 0 + n σ 2 = 25 25 + 70 9 1 25 + 4 9 = 18 . 1 variance: 1 1 σ 2 0 + n σ 2 = 1 1 25 + 4 9 = 2 . 06 2. I want to infer the bias of a coin, using a Bayesian method. My prior on the bias θ is f Θ ( θ ) = ( θ 2 0 θ 1 0 otherwise I throw the coin 10 times, and get 8 heads and 2 tails. (a) What is the MAP estimate for θ ? arg max θ θ 2 10 8 θ 8 (1 - θ ) 2 = arg max θ θ 10 (1 - θ ) 2 differentiate and set to zero: 10 θ 9 (1 - θ ) 2 - 2 θ 10 (1 - θ ) = 0 10 - 10 θ - 2 θ = 0 θ = 5 / 6 (b) Is it an unbiased estimator? No 3. I am interested in seeing whether a sequence of 16 observations has zero mean. My null hypothesis is H 0 : μ = 0, and my alternative is H 1 : μ 6 = 0. I know the variance is 1. What is an appropriate rejection region for the null hypothesis at 0.05 significance? I want to pick γ s.t. P ( ¯ X > γ ) = 0 . 025 under the null. Under the null, ¯ X Normal (0 , 1 / 16), so P ( ¯ X > γ ) = P (4 ¯ X > 4 γ ) = 0 . 025. We have P ( Z > 1 . 96) = 0 . 025, so γ = 1 . 96 / 4 = 0 . 49. So, reject if | ¯ X | > 0 . 49. 4. I set up a motion sensor to look for intruders. The signal when there is no intruder is a normal random variable with mean 0 and variance 1. The signal where there is an intruder is a normal random variable with mean 1 and variance 2. 1
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(a) If I raise the alarm if the signal is greater than 0.7, what is my Type I error? Type I error is probability of false rejection of the null, i.e. P ( X > 0 . 7) under the Normal (0 , 1) null. From tables, we get α = 1 - 0 . 7580 = 0 . 242 (b) What is the corresponding Type II error? Type II error is probability of false acceptance of the null, i.e. P ( X < 0 . 7) under the Normal (1 , 2) null. This is the same as P (( X - 1) /
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