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Module 3C

# Module 3C - IE 361 Module 3 More on Elementary Statistics...

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IE 361 Module 3 More on Elementary Statistics and Metrology Reading: Section 2.2 Statistical Quality Assurance for Engineers (Section 2.2 of Revised SQAME ) Prof. Steve Vardeman and Prof. Max Morris Iowa State University Vardeman and Morris (Iowa State University) IE 361 Module 3 1 / 28

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Basic One and Two Sample Inference Formulas In Module 2 we reminded you that based on a model for y 1 , y 2 , . . . , y n of sampling from a normal distribution with mean μ and standard deviation σ , elementary con°dence limits are y ° t s p n for estimating μ (1) and s s n ± 1 χ 2 upper and s s n ± 1 χ 2 lower for estimating σ (2) (where degrees of freedom for t , χ 2 upper , and χ 2 lower are n ± 1) and then made some simple applications of these limits in contexts that recognize measurement error. Vardeman and Morris (Iowa State University) IE 361 Module 3 2 / 28
Basic One and Two Sample Inference Formulas Parallel to the one sample formulas are the two sample formulas of elementary statistics. These are based on a model that says that y 11 , y 12 , . . . , y 1 n 1 and y 21 , y 22 , . . . , y 2 n 2 are independent samples from normal distributions with respective means μ 1 and μ 2 and respective standard deviations σ 1 and σ 2 . In this context, the so-called "Satterthwaite approximation" gives limits y 1 ± y 2 ° ˆ t s s 2 1 n 1 + s 2 2 n 2 for estimating μ 1 ± μ 2 (3) where appropriate "approximate degrees of freedom" for ˆ t are ˆ ν = ° s 2 1 n 1 + s 2 2 n 2 ± 2 s 4 1 ( n 1 ± 1 ) n 2 1 + s 4 2 ( n 2 ± 1 ) n 2 2 Vardeman and Morris (Iowa State University) IE 361 Module 3 3 / 28

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Basic One and Two Sample Inference Formulas This degrees of freedom formula is one that you may not have seen in an elementary statistics course, where sometimes only methods valid when one assumes that σ 1 = σ 2 are presented. It turns out that above min (( n 1 ± 1 ) , ( n 2 ± 1 )) ² ˆ ν so that a simple conservative version of this method (that you are welcome to employ in IE 361) uses degrees of freedom ˆ ν ³ = min (( n 1 ± 1 ) , ( n 2 ± 1 )) Further, in the two-sample context, standard elementary con°dence limits for comparing standard deviations are s 1 s 2 ´ 1 q F ( n 1 ± 1 ) , ( n 2 ± 1 ) ,upper and s 1 s 2 ´ 1 q F ( n 1 ± 1 ) , ( n 2 ± 1 ) ,lower for σ 1 σ 2 (4) (and be reminded that F ( n 1 ± 1 ) , ( n 2 ± 1 ) ,lower = 1 / F ( n 2 ± 1 ) , ( n 1 ± 1 ) ,upper so that standard F tables giving only upper percentage points can be employed). Vardeman and Morris (Iowa State University) IE 361 Module 3 4 / 28
Basic One and Two Sample Inference Formulas If these formulas do not look familiar, you should immediately stop and review them. Their use can, for example, be seen in Chapter 6 of Vardeman and Jobe±s Basic Engineering Data Collection and Analysis or in the Stat 231 text. Here we will consider a variety of applications of them to problems that arise in metrological studies for quality assurance. Our basic objective is to amply illustrate (and help you develop the thought process necessary to successfully employ) the basic insight that How sources of physical variation interact with a data collection

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