# hw6sol - 36-226 Summer 2010 Homework 6 Solutions 1. (a) You...

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36-226 Summer 2010 Homework 6 Solutions 1. (a) You know that X/σ N(0 , 1) so X 2 2 χ 2 (1). (b) 0 . 95 = P ± χ 2 1 - α/ 2 (1) < X 2 σ 2 < χ 2 α/ 2 (1) ² = P X 2 χ 2 1 - α/ 2 (1) > σ 2 > X 2 χ 2 α/ 2 (1) ! = P ± X 2 0 . 001 > σ 2 > X 2 5 . 0239 ² Therefore [0 . 199 X 2 , 1018 . 26 X 2 ] is a 95% C.I. for σ 2 . (c) 0 . 95 = P ± X 2 σ 2 > χ 2 1 - α (1) ² = P ± σ 2 < X 2 χ 2 1 - α (1) ² = P ± σ 2 < X 2 . 0039 ² Therefore [0 , 254 . 31 X 2 ] is a 95% C.I. for σ 2 . (d) 0 . 95 = P ± X 2 σ 2 < χ 2 α (1) ² = P ± σ 2 > X 2 χ 2 α (1) ² = P ± σ 2 > X 2 3 . 841 ² Therefore [0 . 26 X 2 , ) is a 95% C.I. for σ 2 . 2. (a) You know that X - μ σ 2 / n N(0 , 1) so this means that n ( X - μ ) N(0 , 1). Thus, a 90% CI for μ is given by X ± z 0 . 05 n = X ± 1 . 64 n . (b) As n increases, the interval gets narrower. Thus we have conﬁdence that μ is concen- trated in a shrinking interval around μ .

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## hw6sol - 36-226 Summer 2010 Homework 6 Solutions 1. (a) You...

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