Homework 3 solutions

# Homework 3 solutions - Statistics 5021 Homework 3...

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Statistics 5021 – Homework 3 (solutions) There are 20 total points (1 point for each part of each question and 3 points for handing in the assignment). One question will be graded for correctness. This homework is due Thursday, February 17 in your lab section. 1. Deﬁne the following terms. (a) Random sample Solution: a sequence of random variables X 1 ,...,X n that are independent and all have the same distribution. (b) Realization of a random sample Solution: the sequence of values x 1 ,...,x n that the random sample X 1 ,...,X n took on after the experiment was performed. 2. Suppose that we model the time, in minutes, it takes to drive from Apple Valley to the University of Minnesota during rush hour with the N ( μ = 44 = 8) distribution. In particular, 86 individuals plan to make this drive. Let X 1 ,...,X 86 denote the yet-to-be measured drive times. Assume that X 1 ,...,X 86 are a random sample from N ( μ = 44 = 8). (a) What is the the probability distribution of the sample mean drive time of these 86 individuals who plan to make this drive? (specify the values of its parameters). Solution: Since X 1 ,...,X 86 are a random sample from N ( μ = 44 = 8), using Theorem A, the mean and standard deviation of ¯ X are μ ¯ X = μ = 44 minutes and σ ¯ X = σ/ n = 8 / 86 = 0 . 8627 minutes. Using the ﬁrst part of Theorem B, ¯ X N ( μ ¯ X ¯ X ). (b) What is the name of the probability distribution of the random variable: ¯ X - 44 8 / 86 Solution: We know that ¯ X is normal with mean μ ¯ X = μ = 44 and standard deviation σ ¯ X = σ/ n = 8 / 86. Using the Normal–Standard normal rule, the random variable: ¯ X - μ ¯ X σ ¯ X must be standard normal. 1

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(c) What is the probability that we will observe a sample mean of at least 45 minutes based on this sample of size 86? Solution: Using the fact that ¯ X is normal with mean μ ¯ X = μ = 44 and standard deviation σ ¯ X = σ/ n = 8 / 86: P ( ¯ X 45) = P ± ¯ X - μ ¯ X σ ¯ X > 45 - μ ¯ X σ ¯ X ² = P ± Z > 45 - 44 8 / 86 ² = P ( Z > 1 . 159202) = 1 - P ( Z < 1 . 159202) = 1 - pnorm (1 . 159202) = 0 . 1231869 where Z is standard normal. 2
(d) One can use R to simulate the probability computed in problem 2c. This is done by generating reps = 10 5 realizations of a random sample from N ( μ = 44 = 8), and computing the proportion of these realizations for which the observed sample mean was at least 45 minutes. Run the following R code (you may copy and paste it) to compute the simulated value of P ( ¯ X 45). Compare this value to that computed in problem 2c.

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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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Homework 3 solutions - Statistics 5021 Homework 3...

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