MATH 3200 Exam 3 Dr. Syring

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1 . Suppose n eligible voters are polled (randomly sampled) from a population of size N. The poll asks voters whether they support or do not support increasing local taxes to fund public parks. Let M be the true number of eligible voters supporting the tax policy. Let the responses of the polled voters be recorded as X,..., X n with X i either or 0 for supporting or not supporting the new tax policy. What is an unbiased estimator of M/N? a) n X i b) n n X i c) n n X i d) n n X i e) n+ n X i 2. For a random sample X,..., X n from an unknown population with an unknown, finite mean µ, which of the following are always true of the estimator X? a) X is unbiased for µ. b) X is consistent for µ. c) X is approximately normally distributed. d) Both a) and b). e) All of the above. 3. For independent and identically distributed random variables X,..., X n from a Poisson distribution with parameter λ, which of the following are true? a) E(X i ) = λ, for each i. b) V (X i ) = λ, for each i. c) V ( X) = λ/n. d) n X has a Poisson distribution with parameter nλ. e) All of the above. 4. For independent and identically distributed random variables X,..., X n from an Exponential distribution with parameter θ, which of the following are true?

2 a) V (X i ) = θ 2, for each i. b) Xi, has a Gamma distribution with shape and scale equal to n and θ/n, respectively. c) X, has a Gamma distribution with shape and scale equal to n and θ/n, respectively. d) Both a) and b). e) Both a) and c). 5. For independent and identically distributed random variables X,..., X n from a Poisson distribution with parameter λ, which of the following are unbiased estimators of λ? a) X b) S 2 = n c) S = S 2 d) Both a) and b). e) Both a) and c). n (X i X) 2 6. For independent and identically distributed random variables X,..., X n from a Bernoulli distribution with parameter p, which of the following are true? a) Xi has a Binomial distribution with parameters n and p. b) V (X i ) = np( p), for each i. c) V (X i ) = p( p), for each i. n d) Both a) and b). e) Both b) and c). Solutions:. c 2. d 3. e 4. e 2

3 5. d 6. a 7. Suppose X, X 2,...X n is a random sample from a normal population with mean µ and variance σ 2. Which of the following are unbiased estimators of the population median? a) X = n n X i b) 2 (n) X () ], where X (k) denotes the k th largest observation. c) X i for any i =,..., n. d) Both a) and c). e) All of the above. Solution: d). The population median is equal to the population mean for symmetric distributions. So, either the sample mean or any randomly sampled observation would have expectation eual to the mean, and also the median. 8. Recall the Cauchy distribution with probability density function f(x) =, x, µ R, π( + x 2 ) and E(X k ) = for k =, 2,... Suppose you randomly sample X, X 2,..., X n from a Cauchy population. How would you expect X to behave as an estimator? a) X converges to the population mean by the LLN. b) X has an approximate normal distribution by the CLT. c) X doesn t estimate anything because the population has no finite mean. d) Both a) and b). Solution: c). The LLN requires a finite population mean and the CLT requires a finite population mean and variance. 3

4 9. Suppose X, X 2,..., X n is a random sample from a population with mean µ and variance σ 2. A Math 3200 student uses a calculator to compute an estimate of the population variance. That calculator includes two built-in functions giving the following two estimates: s 2 = n (x i x) 2 s 2 2 = n (x i x) 2. Find the difference in the biases of the two estimators, that is Bias(s 2 ) Bias(s 2 2). a) 0 b) /n c) σ 2 /n d) σ 2 /n Solution: Find Bias(s 2 ) = E(s 2 σ 2 ). E(s 2 ) = E( n (X i X) 2 ) = n E( Xi 2 2X i X + X2 ) = n [ E(Xi 2 ) ne( X 2 )] = n { [σ 2 + µ 2 ] n E(X Xn 2 + X X X n X n )} = n {n(σ2 + µ 2 ) n [n(σ2 + µ 2 ) + n(n )µ 2 ] = n n σ2. 4

5 Hence, the bias is n σ2. It is clear from the above calculation that Bias(s 2 2) = The following 7 observations were randomly sampled from a normal population with unknown mean µ and variance σ 2 = Which of the following is an exact 90% confidence interval for µ? a) (2.8, 6.63) b) (2.54, 6.27) c) (2.46, 6.35) d) (2.77, 6.04) Solution: b) x = ( / 7, / 7).. The following 9 observations were randomly sampled from a normal population with unknown mean µ and unknown variance σ Suppose the following confidence interval based on a normal distribution is proposed for µ, (.26, 0.32). What is the approximate coverage probability of this interval estimate? a) 90% b) 95% c) 80% d) 85% 5

6 Solution: c) The approximate CI is given by ( z α/ / 9, z α/ / 9). Find z α/2.28 which corresponds to α = 0.2 so the coverage is 80%, approximately. 2. Refer to the data in problem. above. Find an exact 90% confidence interval for σ 2. a) (2.3, 8.3) b) (2.08, 7.32) c) (2.00, 0.9) d) (.75, 9.92) Solution: d) The exact CI is given by (n )S 2 (n )S 2 ( χ 2 α/2 (n ), (n )). χ 2 α/2 = ( , ). 3. Consider the following realizations of independent and identically distributed Bernoulli random variables with success probability p: Find an approximate 80% confidence interval using the CLT. a) (0.6, 0.59) b) (0.09, 0.66) c) (0.34, 0.4) d) (0.07, 0.68) 6

7 Solution: a). The CLT-based CI is given by (ˆp + z α/2 ˆp( ˆp)/n, ˆp + z α/2 ˆp( ˆp)/n) For this data, ˆp = 3/8, n = 8, and α = 0.2 which gives z α/2 = Refer to the data in problem 3. above. Find an approximate 90% CI based on the appropriate Binomial distribution. a) [0.25, 0.625] b) (0.25, 0.625) c) [0, 0.5] d) [0.25, 0.875] Solution a). 5. The following 2 observations were randomly sampled from and Exponential distribution with unknown parameter θ Find an approximate 99% confidence interval for θ using the CLT. a) (5.2, 9.08) b) (2.33,.87) c) (.82, 2.38) d) (5.3, 8.89) Solution c). 6. Refer to the data in problem 5. above. Find an approximate 98% CI for θ using the appropriate Gamma distribution. Several quantiles g t for Gamma distributions are provided below: a) (3.08, 2.99) 7

8 shape = scale = shape = 2 scale = shape = 7.0 scale = g.005 g.0 g.025 g.05 g.0 g.90 g.95 g.975 g.99 g b) (2.39, 4.7) c) (2.93, 3.49) d) (3.2, 2.72) Solution: d). 7. The following counts were randomly sampled from a Poisson distribution with parameter λ: Find an approximate 90% confidence interval for λ using the CLT. a) (.23, 3.05) b) (.06, 3.23) c) (0.8, 3.48) d) (.6, 3.3) Solution: a). 8. Refer to the data in problem 7. above. Suppose the interval estimate [.428, 2.75] is produced using an approximate Poisson distribution for the observed data. What is the approximate coverage probability of this interval estimate? a) 80% b) 90% c) 95% 8

9 d) 99% Solution: a). 9. Suppose n observations are randomly sampled from a normal population with unknown mean µ and known variance σ 2. A confidence interval is computed and found to have width of 4. How many more observations must be collected in order to reduce the length of a confidence interval to 2 without changing α? a) n b) 2n c) 3n d) 4n d) n 2 Solution c) 20. Free Response. Consider estimation of p for a random sample of Bernoulli random variables X,..., X n. a) Consider estimators of the form c n X i for c 0. Find Bias(c n X i). Bias(c X i ) = E(c X i ) p = cnp p. b) Compute the variance of this estimator, V (c n X i). V (c X i ) = c 2 np( p). c) Compute the mean squared error of c n X i using your answers to a) and b). MSE(c X i ) = V (c X i ) + Bias 2 (c = c 2 np( p) + (cnp p) 2 9 X i )

10 d) Take the derivative of MSE(c n X i) with respect to c and find the value of c minimizing MSE(c n X i). d dc MSE(c X i ) = 2cnp( p) + 2np(cnp p) c[2np( p) + 2n 2 p 2 ] = 2np 2 c = p p + np e) What choice of c minimizes MSE(c n X i) when p = 0.5? What choice of c yields an unbiased estimator of p? If p = 0.5, 0.5 c = n = n +. If c =, the estimator is unbiased. This coincides with the minimum n MSE estimator only when p =. 0