Estimators, Estimates, and Sampling Distributions Example Problems Solutions

# Estimators, Estimates, and Sampling Distributions Example Problems Solutions

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Unformatted text preview: Estimators, Estimates, and Sampling distributions – Examples (with solutions) 1. Suppose that we use the Normal distribution to model the heights of females in the United states. Let’s assume that this Normal distribution has mean μ = 65 inches and standard deviation σ = 3 inches. (a) If we plan to take a random sample of size n , specify the probability distribution of the sample mean ¯ X , (give the distribution name and specify its parameters) Solution: From Theorem A, μ ¯ X = μ = 65 and σ ¯ X = σ/ √ n = 3 / √ n From the first part of theorem B, since ¯ X is based on a random sample from a Normal distribution, ¯ X ∼ N ( μ ¯ X ,σ ¯ X ). (b) If we plan to take a random sample of size n = 20, what is the probability that we will observe a sample mean within 2 inches of the mean μ = 65 inches. Solution: Using the fact that ¯ X ∼ N ( μ ¯ X ,σ ¯ X ), with μ ¯ X = μ = 65 and σ ¯ X = σ/ √ n = 3 / √ 20, we have: P (63 < ¯ X < 67) = P 63- μ ¯ X σ ¯ X < X- μ ¯ X σ ¯ X < 67- μ ¯ X σ ¯ X = P 63- 65 3 / √ 20 < Z < 67- 65 3 / √ 20 = P (- 2 . 981424 < Z < 2 . 981424) = P ( Z < 2 . 981424)- P ( Z <- 2 . 981424) = pnorm (2 . 981424)- pnorm (- 2 . 981424) = 0 . 9971309 (c) Compute and compare the standard deviation of the sample mean ¯ X for the cases n = 10 and n = 50. Solution: When n = 10 is σ ¯ X = 3 / √ 10 = 0 . 9486833 which is larger than when n = 50, σ ¯ X = 3 / √ 50 = 0 . 4242641 1 (d) Compute and compare the standard deviation of the sample mean based on a random sample of size n = 10 for the cases when the standard deviation of the Normal distribution used to model the heights of females is σ = 3 and σ = 8. Solution: When σ = 3 and n = 10, σ ¯ X = 3 / √ 10 = 0 . 9486833 which is smaller than when σ = 8 and n = 10, σ ¯ X = 8 / √ 10 = 2 . 529822 2. The following observations: x 1 ,x 2 ,...,x 24 are a realization of a random sample X 1 ,X 2 ,...,X 24 from a Normal distribution (generated using R’s random number gen- erator). 56.51047 69.72663 72.32081 64.86753 64.56595 67.53100 66.64958 66.99755 66.58817 63.99538 60.96581 58.72800 60.78952 60.52094 70.59555 74.93141 65.59604 61.40853 63.00884 61.78672 65.56646 70.35814 69.93231 60.92663 (a) The observed sample mean is ¯ x = 65 . 20283, for which quantity is this value an estimate? Solution: The mean μ of the Normal distribution from which x 1 ,x 2 ,...,x 24 are a realization of a random sample. (b) If the actual mean and standard deviation of the distribution from which these data are a realization of a random sample is μ = 65 and σ = 4, specify the probability distribution (its name and the values for its parameters) from which ¯ x = 65 . 20283 was a realization....
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## This note was uploaded on 05/06/2011 for the course STAT 5021 taught by Professor Staff during the Spring '08 term at Minnesota.

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Estimators, Estimates, and Sampling Distributions Example Problems Solutions

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