1. 9/3/2015 IENG 486 Statistical Quality & Process Control 1 IENG 486 - Lecture 11 Hypothesis Tests to Control Charts

2. 9/3/2015 IENG 486 Statistical Quality & Process Control 2 Assignment:  Exam: It was supposed to be a long, difficult exam … I’m assuming that you prepared well … Exam Results … 1 st page of hypothesis tests looks grim.  Reading: CH5: 5.3 (already read 5.1-5.2 & 5.4) Start on CH6: all except 6.3.2 & 6.4  Homework 4: Textbook Problems CH5: 9, 11, 13, 23, and 24

3. 9/3/2015 IENG 486 Statistical Quality & Process Control 3 Process for Statistical Control Of Quality  Removing special causes of variation Hypothesis Tests Ishikawa’s Tools  Managing the process with control charts Process Improvement Process Stabilization Confidence in “When to Act” Reduce Variability Identify Special Causes - Good (Incorporate) Improving Process Capability and Performance Characterize Stable Process Capability Head Off Shifts in Location, Spread Identify Special Causes - Bad (Remove) Continually Improve the System Statistical Quality Control and Improvement Time Center the Process LSL  0 USL

4. 9/3/2015 IENG 486 Statistical Quality & Process Control 4 Moving from Hypothesis Testing to Control Charts  A control chart is like a sideways hypothesis test Detects a shift in the process Heads-off costly errors by detecting trends 00 22 22 00 22 22 2-Sided Hypothesis TestShewhart Control ChartSideways Hypothesis Test CLCL LCL UCL Sample Number

5. 9/3/2015 IENG 486 Statistical Quality & Process Control 5 Test of Hypothesis  A statistical hypothesis is a statement about the value of a parameter from a probability distribution.  Ex. Test of Hypothesis on the Mean Say that a process is in-control if its’ mean is  0. In a test of hypothesis, use a sample of data from the process to see if it has a mean of  0.  Formally stated: H 0 :  =  0 (Process is in-control) H A :  ≠  0 (Process is out-of-control)

6. 9/3/2015 IENG 486 Statistical Quality & Process Control 6 Test of Hypothesis on Mean (Variance Known)  State the Hypothesis H 0 :  =  0 H 1 :  ≠  0  Take random sample from process and compute appropriate test statistic  Pick a Type I Error level (  and find the critical value z  /2  Reject H 0 if |z 0 | > z  /2

7. 9/3/2015 IENG 486 Statistical Quality & Process Control 7 UCL and LCL are Equivalent to the Test of Hypothesis  Reject H 0 if: Case 1: Case 2:  For 3-sigma limits z  /2 = 3

8. 9/3/2015 IENG 486 Statistical Quality & Process Control 8 Two Types of Errors May Occur When Testing a Hypothesis  Type I Error -  Reject H 0 when we shouldn't Analogous to false alarm on control chart, i.e., point lays outside control limits but process is truly in-control  Type II Error -  Fail to reject H 0 when we should Analogous to insensitivity of control chart to problems, i.e., point does not lay outside control limits but process is never-the- less out-of-control

9. 9/3/2015 IENG 486 Statistical Quality & Process Control 9 Choice of Control Limits: Trade-off Between Wide or Narrow Control Limits  Moving limits further from the center line Decreases risk of false alarm, BUT increases risk of insensitivity  Moving limits closer to the center line Decreases risk of insensitivity, BUT increases risk of false alarm Sample x UCL LCL CL Sample x UCL LCL CL Sample x UCL LCL CL

10. 9/3/2015 IENG 486 Statistical Quality & Process Control 10 Consequences of Incorrect Control Limits  NOT GOOD: A control chart that never finds anything wrong with process, but the process produces bad product  NOT GOOD: Too many false alarms destroys the operating personnel’s confidence in the control chart, and they stop using it

11. 9/3/2015 IENG 486 Statistical Quality & Process Control 11 Differences in Viewpoint Between Test of Hypothesis & Control Charts Hypothesis TestControl Chart Checks for the validity of assumptions. (ex.: is the actual process mean what we think it is?) Detect departures from assumed state of statistical control Tests for sustained shift (ex.: have we actually reduced the variation like we think we have?) Detects shifts that are short lived Detects steady drifts Detects trends

12. 9/3/2015 IENG 486 Statistical Quality & Process Control 12 Example: Part Dimension  When process in-control, a dimension is normally distributed with mean 30 and std dev 1. Sample size is 5. Find control limits for an x-bar chart with a false alarm rate of 0.0027. r.v. x - dimension of part r.v. x - sample mean dimension of part

13. 9/3/2015 IENG 486 Statistical Quality & Process Control 13 Distribution of x vs. Distribution of x

14. 9/3/2015 IENG 486 Statistical Quality & Process Control 14 Ex. Part Dimension Cont'd  Find UCL:  The control limits are:

15. 9/3/2015 IENG 486 Statistical Quality & Process Control 15 Ex. Modified Part Limits  Consider an in-control process. A process measurement has mean 30 and std dev 1 and n = 5. Design a control chart with prob. of false alarm = 0.005 If the control limits are not 3-Sigma, they are called "probability limits".