04/19/23IENG 486 Statistical Quality & Process

Control 1

IENG 486 - Lecture 11

Hypothesis Tests to Control Charts

04/19/23 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 … 1st 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

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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 ProcessLSL 0 USL

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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

2

2

0

2

2

2-Sided Hypothesis Test Shewhart Control ChartSideways Hypothesis Test

CL

LCL

UCL

Sample Number

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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: H0: = 0 (Process is in-control) HA: ≠ 0 (Process is out-of-control)

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Test of Hypothesis on Mean (Variance Known)

State the Hypothesis H0: = 0 H1: ≠ 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 H0 if |z0| > z/2

0 00

x

x xz

n

22

z2z2z

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UCL and LCL are Equivalent to the Test of Hypothesis

Reject H0 if:

Case 1:

Case 2:

For 3-sigma limits z/2 = 3

00 2

xz z

n

0x

02

0 2

xz

n

x z n UCL

0x

02

0 2

xz

n

x z n LCL

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Two Types of Errors May Occur When Testing a Hypothesis

Type I Error - Reject H0 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 H0 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

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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

LCLCL

Sample

x UCL

LCL

CL

Sample

x UCL

LCL

CL

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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

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Differences in Viewpoint Between Test of Hypothesis & Control Charts

Hypothesis Test Control 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

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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 ~ 30, 1x N

~ 30, 1 5Xx N n

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Distribution of

individual

measurements :

,

x

N

Distribution of

sample mean :

, x

x

N n

3 xUCL

3 xLCL

CL

Distribution of x vs. Distribution of x

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Ex. Part DimensionCont'd

Find UCL:

The control limits are:

2

0.00135UCL

P zn

2 2

1 0.00135 0.99865UCL

P z zn

2

3 3 3UCL

z UCL nn

0.0027 / 2 0.00135P x UCL

30CL

3 30 3 1 5 31.34UCL n

3 30 3 1 5 28.66LCL n

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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".