HW8Sol - Homework 8 solution 1. a) The fraction of...

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Homework 8 – solution 1. a) The fraction of out-of-spec shafts can be calculated as P ( X > 5 . 03 | μ = 5 . 02) = 1 - F (5 . 03) = 1 - Φ 5 . 03 - 5 . 02 0 . 01 · = 1 - Φ(1) = 0 . 16 b) The false alarm probability is α = 2 P ( ¯ X > μ 0 + Δ | μ = μ 0 ) = 2(1 - F ( μ 0 + Δ)) = 2 ˆ 1 - Φ ˆ μ 0 + Δ - μ 0 σ/ 5 !! = 2 ˆ 1 - Φ ˆ i 5 2 !! , where μ 0 = 5 . 00 is the nominal diameter, and i = 1 , 2 , 3 ... The probability of failing to detect a shift at a checkpoint can be calcu- lated as β = P ( μ 0 - Δ < ¯ X < μ 0 + Δ | μ = μ 0 + 2 σ ) = F ( μ 0 + Δ) - F ( μ 0 - Δ) = Φ ˆ μ 0 + Δ - μ 0 - 2 σ σ/ 5 ! - Φ ˆ μ 0 - Δ - μ 0 - 2 σ σ/ 5 ! = Φ ±± i 2 - 2 5 - Φ ±± - i 2 - 2 5 , Now, we need to find the long-term expected cost. Let us consider a long time T . During this time, there will be T/ 2 checkpoints. Since it takes on average 1 1 - β checkpoints in order to detect a shift once it happened, and there are on average 25 checkpoints per shift, the fraction of all checkpoints when
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This note was uploaded on 03/17/2008 for the course IE 121 taught by Professor Perevalov during the Spring '08 term at Lehigh University .

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HW8Sol - Homework 8 solution 1. a) The fraction of...

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