11/18/2015 IENG 486 Statistical Quality & Process Control 1 IENG Lecture 07 Comparison of Location (Means)

11/18/2015 IENG 486 Statistical Quality & Process Control2 Assignment:  Preparation: Print Hypothesis Test Tables from Materials page Have this available in class …or exam!  Reading: Chapter 4: through 4.3.4; (skip 4.3.5); through 4.4.3; (skip rest)  HW 2: CH 4: # 1a,b; 5a,c; 9a,c,f; 11a,b,d,g; 17a,b; 18, 21a,c; 22* *uses Fig.4.7, p. 126

11/18/2015 IENG 486 Statistical Quality & Process Control3 Comparison of Means The first types of comparison are those that compare the location of two distributions. To do this: Compare the difference in the mean values for the two distributions, and check to see if the magnitude of their difference is sufficiently large relative to the amount of variation in the distributions Which type of test statistic we use depends on what is known about the process(es), and how efficient we can be with our collected data Definitely Different Probably Different Probably NOT Different Definitely NOT Different

11/18/2015 IENG 486 Statistical Quality & Process Control4 Situation I: Means Test, Both  0 and  0 Known  Used with: an existing process with good deal of data showing the variation and location are stable  Procedure: use the the z-statistic to compare sample mean with population mean  0 (adjust for any safety factor  0 )

11/18/2015 IENG 486 Statistical Quality & Process Control5 Situation II: Means Test  (s) Known and  (s) Unknown  Used when: the means from two existing processes may differ, but the variation of the two processes is stable, so we can estimate the population variances pretty closely.  Procedure: use the the z-statistic to compare both sample means (adjust for any safety factor  0 )

11/18/2015 IENG 486 Statistical Quality & Process Control6 Situation III: Means Test Unknown  (s) and Known  0  Used when: have good control over the center of the distribution, but the variation changed from time to time  Procedure: use the the t-statistic to compare both sample means (adjust for any safety factor  0 ) v = n – 1 degrees of freedom

11/18/2015 IENG 486 Statistical Quality & Process Control7 Situation IV: Means Test Unknown  (s) and  0, Similar S 2  Used when: logical case for similar variances, but no real "history" with either process distribution (means & variances)  Procedure: use the the t-statistic to compare using pooled S, v = n 1 + n 2 – 2 degrees of freedom

11/18/2015 IENG 486 Statistical Quality & Process Control8 Situation V: Means Test Unknown  and  0, Dissimilar S 2  Used when: worst case data efficiency - no real "history" with either process distribution (means & variances)  Procedure: use the the t-statistic to compare, degrees of freedom given by:

11/18/2015 IENG 486 Statistical Quality & Process Control9 Situation VI: Means Test Paired but Unknown  (s)  Used when: exact same sample work piece could be run through both processes, eliminating material variation  Procedure: define variable (d) for the difference in test value pairs (d i = x 1i - x 2i ) observed on i th sample, v = n - 1 dof

11/18/2015 IENG 486 Statistical Quality & Process Control10 Table for Means Comparisons  Decision on which test to use is based on answering (at least some of) the following: Do we know the population variance (  2 ) or should we estimate it by the sample variance (s 2 ) ? Do we know the theoretical mean (  ), or should we estimate it by the sample mean (y) ? Do we know if the samples have equal-variance (  1 2 =  2 2 ) ? Have we conducted a paired comparison? What are we trying to decide (alternate hypothesis)?

11/18/2015 IENG 486 Statistical Quality & Process Control11 Table for Means 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  Six major test statistics for mean comparisons Two sampling distributions Six confidence intervals Twelve alternate hypotheses

11/18/2015 IENG 486 Statistical Quality & Process Control12 Ex. Surface Roughness  Surface roughness is normally distributed with mean 125 and std dev of 5. The specification is 125 ± and we have calculated that 98% of parts are within specs during usual production.  My supplier of these parts has sent me a large shipment. I take a random sample of 10 parts. The sample average roughness is 134 which is within specifications.  Test the hypothesis that the lot roughness is higher than specifications.

11/18/2015 IENG 486 Statistical Quality & Process Control13 ex. cont'd Draw the distributions for the surface roughness and sample average 134

11/18/2015 IENG 486 Statistical Quality & Process Control14 e.g. Surface Roughness Cont'd  Find the probability that the sample with average 134 comes from a population with mean 125 and std dev of 5.  Should I accept this shipment?

11/18/2015 IENG 486 Statistical Quality & Process Control15 e.g. Surface Roughness Cont'd  For future shipments, suggest good cutoff values for the sample average, i.e., accept shipment if average of 10 observations is between what and what?  We know that encompasses over 99% of the probability mass of the distribution for