Engineering Management 385

Statistical Process Control

Stephen A. Raper, Ph.D., Associate Professor

Chapter 3 – Inferences About Process Quality Continued – Single Sample Case

Objectives

Continue illustration of Statistical Continue illustration of Statistical Inferences for Single Sample CasesInferences for Single Sample Cases

Illustrate case of Inference on the mean of a Illustrate case of Inference on the mean of a normal distribution, variance unknownnormal distribution, variance unknown

Illustrate case of Inference on the Variance Illustrate case of Inference on the Variance of a Normal Distributionof a Normal Distribution

Inference on Population ProportionInference on Population Proportion Work Example ProblemsWork Example Problems

These PowerPoint slides are the author/text These PowerPoint slides are the author/text provided slides. Slight modifications may provided slides. Slight modifications may have occurred and the background used is have occurred and the background used is that required by UMR. The slides generally that required by UMR. The slides generally follow the order in each chapter.follow the order in each chapter.

3-3.4 Inference on the Variance of a Normal Distribution

Hypothesis TestingHypothesis Testing

Hypotheses: HHypotheses: H00: : HH11::

Test Statistic:Test Statistic:

Significance Level, Significance Level, Rejection Region: Rejection Region:

20

2 20

2

20

220

S)1n(

2

1n,2

1

20

2

1n,2

20 or

3-3.4 Inference on the Variance of a Normal Distribution

Confidence Interval on the VarianceConfidence Interval on the Variance Two-Sided:Two-Sided:

See the text for the one-sided confidence See the text for the one-sided confidence intervals.intervals.

1s)1n(s)1n(

P2

1n,2/1

22

21n,2/

2

3-3.5 Inference on a Population ProportionHypothesis TestingHypothesis Testing Hypotheses: HHypotheses: H00: p = p: p = p0 0 HH11: p : p p p00

Test Statistic:Test Statistic:

Significance Level, Significance Level, Rejection Region: Rejection Region:

0

00

0

0

00

0

0

npx)p1(np

np)5.0x(

npx)p1(np

np)5.0x(

Z

2/0 ZZ

3-3.5 Inference on a Population Proportion

Confidence Interval on the Population ProportionConfidence Interval on the Population Proportion Two-Sided:Two-Sided:

See the text for the one-sided confidence See the text for the one-sided confidence intervals.intervals.

1

n

)p̂1(p̂Zp̂p

n

)p̂1(p̂Zp̂P 2/2/

3-3.6 The Probability of Type II Error

Calculation of P(Type II Error)Calculation of P(Type II Error)

Assume the test of interest is HAssume the test of interest is H00: : HH11::

P(Type II Error) is found to beP(Type II Error) is found to be

The Power of the test is then 1 - The Power of the test is then 1 -

o o

n

Zn

Z22

3-3.6 The Probability of Type II Error

Operating Characteristic (OC) CurvesOperating Characteristic (OC) Curves

Operating Characteristic (OC) curve is a graph Operating Characteristic (OC) curve is a graph representing the relationship between representing the relationship between , , , , and and nn. .

OC curves are useful in determining how large a OC curves are useful in determining how large a sample is required to detect a specified sample is required to detect a specified difference with a particular probability.difference with a particular probability.

3-3.6 The Probability of Type II Error

Operating Characteristic (OC) CurvesOperating Characteristic (OC) Curves

3-3.7 Probability Plotting

Probability plotting is a graphical method for Probability plotting is a graphical method for determining whether sample data conform to a determining whether sample data conform to a hypothesized distribution based on a subjective hypothesized distribution based on a subjective visual examination of the data.visual examination of the data.

Probability plotting uses special graph paper Probability plotting uses special graph paper known as known as probability paperprobability paper. Probability paper . Probability paper is available for the normal, lognormal, and is available for the normal, lognormal, and Weibull distributions among others. Weibull distributions among others.

Can also use the computer.Can also use the computer.

3-3.7 Probability Plotting

Example 3-8Example 3-8

j x(j) (j – 0.5)/10 1 1176 0.05 2 1183 0.15 3 1185 0.25 4 1190 0.35 5 1191 0.45 6 1192 0.55 7 1201 0.65 8 1205 0.75 9 1214 0.85 10 1220 0.95

1150 1160 1170 1180 1190 1200 1210 1220 1230 1240

1

5

10

20

3040

506070

80

90

95

99

Data

Per

cent

Normal Probability Plot for Life

ML Estimates

Mean:

StDev:

1195.7

13.3120

Program Completed

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University of Missouri-Rolla