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ISE507 Lecture 6

# ISE507 Lecture 6 - Six Sigma Lecture 6 ANALYZE PHASE 1 2...

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1 Six Sigma Lecture 6 ANALYZE PHASE

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3 Analyze Phase Road Map n Activities n Identify Potential Root Causes n Reduce List of Potential Root Causes n Confirm Root Cause to Output Relationship n Estimate Impact of Root Causes on Key Outputs n Tools n Cause and Effect Analysis n FMEA n Hypothesis Tests/Conf. Intervals n Simple and Multiple Regression n ANOVA n Components of Variation

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4 Hypothesis Testing
5 Normal Distribution n Variable X ~ Normal ( μ , σ 2 ) n Probability=area under the density function curve μ

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6 Excel Functions μ t μ b a n P(x≤t) = NORMDIST(t, μ,σ ,true) n t=NORMINV(probability, μ,σ29 n P(a<x≤b) =P(x≤b)-P(x≤a) =NORMDIST(b, μ,σ ,true)- NORMDIST(a, μ,σ ,true)
7 Desirable features of Normal Distribution n If X~Normal( μ,σ 2 ), then Z=(x- μ )/ σ ~ Normal(0,1) n This is the Standard Normal Distribution n Claim: Every problem on an arbitrary normal distribution can be transferred to a problem on the Standard Normal Distribution 0 z P(Z≤z)

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8 Translation to Z = Normal(0,1) For an arbitrary X~Normal( μ,σ 2 ): μ t μ b a n P(a≤X≤b) = P[(a- μ )/ σ ≤ (X- μ )/ σ ≤ (b- μ )/ σ ] = P[(a- μ )/ σ ≤ Z ≤ (b- μ )/ σ ] n P(X≤t) = P[(X- μ )/ σ ≤ (t- μ )/ σ ] = P[Z≤(t- μ )/ σ ] Example: Suppose X~Normal(3,25) P(X≤5) = ? P(-2≤X≤5)
9 n Usually μ and σ 2 are unknown, we try to estimate them from random samples n Suppose n=sample size n Let xi=outcome of the ith item in the sample n Assumptions: l E(xi)= μ ; var(xi)= σ 2 l The outcomes of the ith item (xi) and the jth item (xj) are independent l Xbar=(x1+x2+…+xn)/n is the sample average l Then, by properties of random variables E(Xbar)= μ and var(Xbar)= σ 2 /n (Central Limit Theorem) Sampling Distribution

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10 Z α/2 0 α/2 α/2 Confidence Interval for Population Mean n Case 1: σ is known l The (1- α
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