Midterm Examination. Mth 136 = Sta 114. Wednesday, 2000 March 8, 2:20 3:35 pm

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1 Midterm Examination Mth 136 = Sta 114 Wednesday, 2000 March 8, 2:20 3:35 pm This is a closed-book examination so please do not refer to your notes, the text, or to any other books. You may use a two-sided single sheet of your own notes, if you wish, but you may not share materials. A normal distribution table, the PDF handout, and a blank worksheet are attached to the exam. If you don t understand something in one of the questions feel free to ask me, but please do not talk to each other. You may use a calculator but not a laptop computer. You must show your work to get partial credit. Unsupported answers are not acceptable, even if they are correct. Please give all numerical answers as fractions in lowest terms (simplify!) or as decimals correct to four places. You should spend about minutes on each problem. It is to your advantage to write your solutions as clearly as possible, since I cannot give credit for solutions I do not understand. Good luck. Cheating on exams is a breach of trust with classmates and faculty, and will not be tolerated. After completing the exam please acknowledge the Duke Honor Code: I have neither given nor received any unauthorized aid on this exam. Signature: 1. /20 2. /20 3. /20 4. /20 5. /20 Total: /100

2 Problem 1: Suppose that the proportion θ of defective items in a large shipment is unknown, and that the prior distribution of θ is a beta distribution with parameters α = 2 and β = 200. If n = 100 items are selected at random for inspection, and if three of these items are found to be defective, a) (10) What is the posterior distribution for θ? b) (5) The posterior mean of θ is also the conditional probability, given these data, that the next (i.e. 101 st) item will be defective. Evaluate the posterior mean: c) (5) Write down an expression for computing the probability that θ is less than.01 in any one (your choice) of the following forms: an integral, or a command in S-Plus, Mathematica, Maple, or Matlab. Do you expect it to be more or less than 1 2? Spring April 7, 2000

3 Problem 2: The lengths X j of a number of butterfly wings are measured and recorded; we view these as independent normally-distributed values with some unknown mean θ but known variance σ 2 = 0.50, hence with pdf f(x θ) = 1 e (x θ)2. π We observe n = 2 butterflies, with wing lengths x 1 = 5 and x 2 = 11. a) (5) For these data, find the Likelihood Function f n (x θ); b) (5) Find the maximum likelihood estimate ˆθ n (x): c) (5) With a uniform prior p(θ) 1, find the posterior density function p(θ x): d) (5) Use this prior to give a 90% equal-tail posterior credible interval for θ (remember that there is a normal table attached to this test). Spring April 7, 2000

4 Problem 3: Survey participants are often reluctant to answer embarassing questions honestly questions about illegal drug use, for example. One way investigators overcome this is by asking questions in the following way: Please toss a fair coin. If it shows Heads, answer the question below; if it shows Tails, please check Yes : Y N Have you used cocaine within the past 12 months? Let θ be the proportion of the population who have used cocaine in the past twelve months, and let y be the number of Yes answers among n subjects who are asked the question above. a) (8) Find the likelihood function f n (y θ). b) (4) Find the maximum likelihood estimate ˆθ n. Be careful (see c. below). c) (4) If 45 of 100 subjects answer Yes, what is the estimate? Is it reasonable? d) (4) Write down an expression for the posterior mean E[θ y] for a uniform prior density p(θ) = 1. Spring April 7, 2000

5 Problem 4: Tracy and Billy are trying to estimate the height θ of Duke Chapel, using triangulation and parallax and lots of other impressivesounding words. They take n = 9 independent measurements {x i } and find x n = 1 n xi = A wise and venerable surveyor happens to pass by during their efforts, and explains to them that measurements of the kind they are performing are subject to normally-distributed errors, always (this is the cool part) with exact known variance of σ 2 = Thus x i No(θ, 2.25). a) (5) Tracy says they should base their confidence intervals on this Normal Distribution, with σ 2 = 2.25; Billy (who likes Irish bars and Guinness Stout) says that they should compute Sn 2 = 1 (xi x n 1 n ) 2 and use the t distribution with ν = 8 degrees of freedom for their intervals. Who is right? b) (5) Tracy s interval (see above) is [63.36, 64.64]. What confidence level did s/he use? Was it 80%, 90%, 95%, 99%, or something else? How do you know? c) (5) What is the probability P[θ [63.36, 64.64] θ], for Tracy s interval? Does the question make sense? d) (5) Chris thought that the chapel was about 60m high, so used a No(60, 10 2 ) prior distribution to compute a 95% Bayesian credible interval for θ; s/he found m 1 = [9/2.25] [1/100] = [9/2.25] + [1/100] τ1 2 = 1/[9/ /100] = = and reported m 1 ±1.96τ 1 = [63.02, 64.98]. What is the probability P[θ [63.02, 64.98] x], for Chris interval? Does the question make sense? Spring April 7, 2000

6 Problem 5: A hat contains n = 10 coins, eight of which are fair (so that P[H] = 1/2) and two of which are biased with P[H] = 3/4. A single coin is drawn at random from the hat. All questions below are about this one coin; it is not replaced, and no other coin is drawn. Be sure to simplify your answers below. a) (5) On the first toss, it lands Heads. What is the probability that it is a biased coin? b) (5) It is tossed a second time. What is the (conditional) probability that it will land Heads this time too, given Heads on the first toss? c) (5) If it does land Heads on both the first and second tosses, what is the probability that it is a biased coin? d) (5) Denote by θ the probability of heads for this particular coin, and suppose that it lands Heads on each of the first n tosses. Give the conditional probability distribution for θ, for each n (Hint: First decide if θ has a discrete or a continuous conditional distribution). Simplify! Spring April 7, 2000

7 Extra worksheet, if needed: Spring April 7, 2000

8 ... Name: Φ(x) = x 1 2π e t2 /2 dt: x Table 5.1Area Φ(x) under the Standard Normal Curve to the left of x. x

Practice Final Examination

Practice Final Examination Mth 136 = Sta 114 Wednesday, 2000 April 26, 2:20 3:00 pm This is a closed-book examination so please do not refer to your notes, the text, or to any other books You may use a