IENG 486 Statistical Quality & Process Control 11/18/2018 IENG 486 - Lecture 07b Comparison of Spread (Variances) & Operating Characteristic Curves 11/18/2018 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009 D.H. Jensen & R. C. Wurl

IENG 486 Statistical Quality & Process Control 11/18/2018 Assignment: Preparation: Print Hypothesis Test Tables from Materials page Have this available in class …or exam! Reading: Chapter 4: 4.1.1 through 4.3.4; (skip 4.3.5); 4.3.6 through 4.4.3; (skip rest) HW 2: CH 4: # 1, 5, 9, 11, 17, 18, 21, 22* *uses Fig.4.7, p. 126 11/18/2018 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009 D.H. Jensen & R. C. Wurl

Comparison of Variances The second types of comparison are those that compare the spread of two distributions. To do this: Compute the ratio of the two variances, and then compare the ratio to one of two known distributions as a check to see if the magnitude of that ratio is sufficiently unlikely for the distribution. Assumption that the data come from Normal distributions is very important. Assess how normally data are distributed prior to conducting either test. Definitely Different Probably Different Probably NOT Different Definitely NOT Different 11/18/2018 IENG 486 Statistical Quality & Process Control

Situation VII: Variance Test With 0 Known Used when: existing comparison process has been operating without much change in variation for a long time Procedure: form ratio of a sample variance (t-distribution variable) to a population variance (Normal distribution variable), v = n - 1 degrees of freedom 11/18/2018 IENG 486 Statistical Quality & Process Control

Situation VIII: Variance Test With 0 Unknown Used when: worst case variation comparison process for when there is not have enough prior history Procedure: form ratio of a sample variances (two 2-distributions), v1 = n1 – 1 degrees freedom for numerator, and v2 = n2 – 1 degrees freedom for the denominator Note: 11/18/2018 IENG 486 Statistical Quality & Process Control

Table for Variance Comparisons Decision on which test to use is based on answering the following: Do we know the theoretical variance (s2) or should we estimate it by the sample variances (s2) ? What are we trying to decide (alternate hypothesis)? 11/18/2018 IENG 486 Statistical Quality & Process Control

Table for Variance Comparisons These questions tell us: What sampling distribution to use What test statistic(s) to use What criteria to use How to construct the confidence interval Four primary test statistics for variance comparisons Two sampling distributions Two confidence intervals Six alternate hypotheses Table construction Note: F1-a, v1, v2 = 1 Fa, v2, v1 11/18/2018 IENG 486 Statistical Quality & Process Control

Operating Characteristic (OC) Curve IENG 486 Statistical Quality & Process Control 11/18/2018 Operating Characteristic (OC) Curve Relates the size of the test difference (d) to the Type II Error (b) for a given risk of Type I Error (a) Ex: for 2-sided z-statistic Designing an experiment involves a trade-off in sample size versus power of the test to detect a difference (effect) The greater the difference in means (d), the smaller the chance of Type II Error for a given sample size and a. As the sample size increases, the chance of Type II Error decreases for a specified a and given difference in means (d). 11/18/2018 IENG 486 Statistical Quality & Process Control (c) 2002 - 2009 D.H. Jensen & R. C. Wurl

Operating Characteristic Curve 11/18/2018 IENG 486 Statistical Quality & Process Control

IENG 486 Statistical Quality & Process Control O.C. Curve Use Agree on acceptable a Estimate anticipated d and s to compute d: d = | m1 - m2| = |d| s s Look for where d intersects with desired b (Probability of keeping H0 when H0 is false) to estimate required sample size (n) 11/18/2018 IENG 486 Statistical Quality & Process Control

OC Curve Example (uses Fig 4.7, p.126) Assume our previous problem had a process std. dev. of 18 (instead of 5), and the same means (125 population & spec, 134 supplier sample). Assume the boss wants  = 0.05 of exceeding either the high or low spec. for such a sample. Probability of what (in English)? Assume supplier needs  = 0.2 What sample size is needed to fit these constraints? 11/18/2018 IENG 486 Statistical Quality & Process Control

Operating Characteristic Curve β = 0.2 d = 0.5 11/18/2018 IENG 486 Statistical Quality & Process Control