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Chapter 9 : Some Other Points about Normal Distributions A: Prediction intervals Example 1 : The daily sales volume of a small department store has a N(87, 5²) distribution where the monetary unit is in \$1000. Predict, with 95% probability, tomorrow’s sales volume.

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Solution: For any normal distribution N(μ,σ²) we have from Table B So that a 95% prediction interval for x is With the data that μ=87 and σ=5, we have Interpretation : There is a 95% chance that the sales volume will be between 77.2 and 96.8 thousand dollars tomorrow. σ μ 96 . 1 ) ( 95 . 0 = x PI ) 96 . 1 96 . 1 ( 95 . 0 σ μ σ μ + < < - = x P ) 8 . 96 , 2 . 77 ( 5 96 . 1 87 ) ( 95 . 0 = × = x PI
Common prediction intervals B: The Six-Sigma Criterion “Six-sigma” is nowadays a rather popular term talked about by managers, quality control engineers and business people. Some authors have written books on it. The basic idea is very simple. Suppose you are a manufacturer. You are concerned about whether the quality of your product can meet customer’s specifications. σ μ σ μ σ μ 576 . 2 ) ( 960 . 1 ) ( 645 . 1 ) ( 99 . 0 95 . 0 90 . 0 = = = x PI x PI x PI

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Example 2 Manufacturer A is considering signing a contract with customer B to supply him with an electronic device whose resistance is of key concern. B’s requirements are: Resistance = 72.5 ± 2.5Ω (ohms) i.e. Upper specification limit = USL = 75Ω, Lower specification limit = LSL = 70Ω. A has checked his production conditions and found that his product has a N(72.5, 0.3²) Should A accept the deal?
Solution : Producer’s capability is: Resistance=μ±6σ, i.e. Upper specification limit = UPL = 74.3Ω, Lower specification limit = LPL = 70.7Ω. These production limits are well within B’s specification limits, therefore A can comfortably accept the deal.

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Note : When the producer’s six-sigma production limits fall outside the customer’s specification limits, he should try hard to improve his quality control procedure, particularly in reducing his σ-value. Various people have written books on 6σ- management. 1. Snee, R.D; & Hoerl, R.W. (2003), “Leading Six Sigma”, New York Financial Times, Prentice-Hall 2. Chowdhury, S. (2002) “ Design for Six-sigma”, New York Financial Times, Prentice-Hall
C : N(0,1) The Standard Normal Distribution Of all normal distributions N(μ,σ²), the one with μ=0 and σ=1 i.e., the N(0,1)-distribution, occupies a special position and has been tabulated (in detail) as in Tables A and B. When we have a general normal distribution X ~ N(μ,σ²) we often standardize it (or find its S/S ratio) as then by the formulae: σ μ - = x z

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Mean(A+Cx) = A + CMean(x) and s.d.(A+C x)=|C|s.d.(x), we have that Mean (z)=0 and S.D.
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