ps06 - Statistics 403 Problem Set 6 Due in lab on Friday...

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Unformatted text preview: Statistics 403 Problem Set 6 Due in lab on Friday, October 29th 1. Three processes for collecting data X 1 ,...,X 10 are described below. For each, deter- mine the 15 th , 50 th , and 90 th percentiles of the sampling distribution of ¯ X . (a) The data are collected as independent and identically distributed values from a distribution with mean 2 and standard deviation 0 . 9. Solution: The quantile is a solution to an equation of the following form: P ( ¯ X ≤ Q ) = 0 . 15 (for the 15 th percentile). To solve the equation you need to standardize ¯ X , which will require that you have the expected value E ¯ X = 2 and the variance var( ¯ X ) = 0 . 9 2 / 10. For the 15 th percentile: . 15 = P ( ¯ X ≤ Q ) = P ( Z ≤ √ 10( Q- 2) / . 9) , then solve √ 10( Q- 2) / . 9 =- 1 . 04 to get Q = 2- 1 . 04 · . 9 / √ 10 ≈ 1 . 7 . The other quantiles are Q = 2 (for the 50 th percentile) and Q = 2 + 1 . 28 · . 9 / √ 10 ≈ 2 . 36 (for the 90 th percentile). (b) The data are collected as independent values. The i th data value X i is collected from a distribution with mean 2 and standard deviation 0 . 7 (if i ≤ 5) or 1 . 1 (if i > 5). Solution: Again we have E ¯ X = 2, but to get the variance of ¯ X , we need to use the covariance matrix 1 Σ = . 7 2 ··· ··· . 7 2 ··· ··· . 7 2 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 1 . 1 2 ··· ··· 1 . 1 2 ....
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ps06 - Statistics 403 Problem Set 6 Due in lab on Friday...

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