homework6_solutions - Math 1011 Intro to Statistics Spring...

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Math 1011– Intro to Statistics Spring 2011 HOMEWORK 6 Due Monday, 3/28/11 1. A box contains slips of paper with the numbers 2, 9,10, 11. Pieces of paper are drawn from the box with replacement . (a) Find the expected value and standard error of the average of one number drawn from the box. (b) Find the expected value and standard error of the average of 25 numbers drawn from the box. (c) Find the expected value and standard error of the average of 100 numbers drawn from the box. Solution: (a) Let X be the random variable of drawing one number from the box. Then E ( X ) = 2( 1 4 ) + 9( 1 4 ) + 10( 1 4 ) + 11( 1 4 ) = 8 SE ( X ) = radicalbigg (2 - 8) 2 ( 1 4 ) + (9 - 8) 2 ( 1 4 ) + (10 - 8) 2 ( 1 4 ) + (11 - 8) 2 ( 1 4 ) = 5 2 3 . 53 (b) E ( X 1 + ... + X 25 25 ) = E ( X ) = 8 SE ( X 1 + ... + X 25 25 ) = 1 25 E ( X ) = 1 2 0 . 71 (c) E ( X 1 + ... + X 100 100 ) = E ( X ) = 8 SE ( X 1 + ... + X 100 100 ) = 1 100 E ( X ) = 1 2 2 0 . 35 2. Consider the box of numbers from the first problem. Estimate the following probabilities: (a) The probability that the average of 25 draws is larger than 9 (b) The probability that the average of 25 draws is smaller than 7.5 Solution:
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