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Throughout Don Benbow's extensive career, teaching mathematics and statistical courses to college students and corporate employees, one common question always popped up: how do you fix statistical errors in quality? Most textbooks and courses tend to emphasize how to perform statistical analysis and give little attention to errors that can occur in the process. This book intends to teach readers how to avoid common pitfalls by providing examples and scenarios based on similar real-world events. The book also provides caveats (or quick tips) to help readers navigate their way through statistical methodology.

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Sciences & Techniques

Statistiques

Auditing

Economie gestion

Economics

Business

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Publié par	ASQ Quality Press
Date de parution	01 juin 2021
Nombre de lectures	0
EAN13	9781636940014
Langue	English
Poids de l'ouvrage	1 Mo

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Mistakes in Quality Statistics
and How to Fix Them
Donald W. Benbow
ASQ Quality Press
Milwaukee, Wisconsin
Mistakes in Quality Statistics and How to Fix Them
© 2021 by Donald W. Benbow
All rights reserved. Published 2021
Publisher’s Cataloging-in-Publication data
Names: Benbow, Donald W., 1936–, author.
Title: Mistakes in quality statistics and how to fix them / by Donald W. Benbow.
Description: Includes bibliographical references. | Milwaukee, WI: Quality Press, 2021.
Identifiers: LCCN: 2021935792 | ISBN: 978-1-63694-000-7 (paperback) | 978-1-63694-001-4 (epub)
Subjects: LCSH Statistics. | Statistics—Evaluation. | Statistics—Methodology. | Auditing. | Quality control. | BISAC BUSINESS & ECONOMICS / Auditing | MATHEMATICS / Probability & Statistics / General
Classification: LCC QA276 .B46 2021| DDC 519.5—dc23
No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
ASQ advances individual, organizational, and community excellence worldwide through learning, quality improvement, and knowledge exchange.
Attention bookstores, wholesalers, schools, and corporations: Quality Press books are available at quality discounts with bulk purchases for business, trade, or educational uses. For information, please contact Quality Press at 800-248-1946 or books@asq.org.
To place orders or browse the selection of ASQ Excellence and Quality Press titles, visit our website at http://www.asq.org/quality-press .
Table of Contents
Cover
Title page
Copyright
Preface
Chapter 1. Hypothesis Tests
1.1 Hypothesis Test for Two Population Means
1.2 Hypothesis Test for a Population Proportion
1.3 Hypothesis Test for Population Standard Deviation
Chapter 2. Correlation and Causation
2.1 Lurking Variable
2.2 Correlation or Causation?
2.3 Correlation and Prediction
Chapter 3. Margin of Error and Confidence Intervals
3.1 Margin of Error and Confidence Intervals for Population Means
3.2 Hypothesis Test for Two Population Means
3.3 Margin of Error and Confidence Intervals for Proportions
Chapter 4. Control Charts
4.1 Pre-control Charts
4.2 Calculations for a Process with a Normal Population
Chapter 5. Measurement Systems
5.1 Measurement System Variation
5.2 Evaluating a Measuring System for Continuous Data
5.3 Sample Procedure for Measurement System Analysis
5.4 Diagnosis Using the GRR
5.5 Stability
5.6 Measurement Systems and Capability Analysis
Chapter 6. Designed Experiments
6.1 Main Effects
6.2 Interaction Effects
6.3 Fractional Factorial Designs
6.4 Balanced Designs
Chapter 7. Sampling Plans
7.1 Attribute Sampling Plans
7.2 Double Sampling Plans
7.3 Evaluating Sampling Plans
7.4 Variables Sampling Plans
Chapter 8. Probability
8.1 Summary of Rules
Chapter 9. Some General Caveats
Appendix
Notes
Further Study
Recommended Reading
About the Author
Preface
This book began as a presentation titled “Caveats Regarding the Use of Statistics” that I made during the May 2009 World Quality Conference. It is not intended to serve as a statistics textbook. In fact, it assumes the reader has some familiarity with the subject, perhaps through an introductory course. Texts and courses tend to emphasize how to perform statistical analysis and give little attention to errors that can occur in the process. The purpose of this book is to show readers how to avoid pitfalls. The examples and case studies, although based on similar events, do not include actual data.
1
Hypothesis Tests
Inferential statistics uses analysis of samples from a population to generate conclusions about that population. It provides techniques for studying variation that assist in quality improvement. This chapter uses examples of hypothesis testing as a standard aid for decision-making.
FOR EXAMPLE:
Example 1.1
A quality improvement team has been asked whether Method A or Method B provides a higher mean value for the response R.
(Methods A and B could be two different advertising methods, two different vat agitation rates, two different oven temperatures, etc.)
The team uses Method A and Method B once each and finds that the resulting r -value is higher for Method B. The team prepares a report with the graph shown in Figure 1.1 .

FOR EXAMPLE:
Example 1.2
Example 1.2 is the same scenario as Example 1.1, but the team decides to use the two methods five times each. The results are shown in Figure 1.2 .

The team recognizes that variation exists in the data. They realize that to make a reasonable estimate of the mean value for each method, they will need many more values.

CAVEAT 1.1
It is important to seek out variation so it can be analyzed.
The team conducts 35 repetitions of Method A and 32 of Method B. The resulting data are graphed in Figure 1.3 .
The graph in Figure 1.3 shows the samples from the two populations. The mean of the population made by Method A appears to be larger than the mean of the population made by Method B.

CAVEAT 1.2
Always review the data collection process to look for data entry, coding, and editing errors. Statisticians also speak of sampling error, which is the unavoidable error that occurs anytime sampling is used as a basis for a decision about the population from which the sample was drawn. Sampling error can be reduced by using a larger sample.
The team calculates the mean and standard deviation of each sample and gets the following results:

These sample statistics are estimates for the population parameters µ A , σ A , µ B , and σ B , respectively.
1.1 HYPOTHESIS TEST FOR TWO POPULATION MEANS
Because of the amount of variation and the relative closeness of the two sample means, the team decides to use a hypothesis test for two population means, which will guide them in reaching their recommendation. They proceed through textbook steps for this test:
Verify that the conditions for using the test have been met.
The samples are independent.
The populations are normally distributed or each sample size ≥ 30.
Determine the significance level. In this case, the team decides on 0.05.
State the null hypothesis. In this case, it is H o : μ A = μ B . The alternative hypothesis is H a : μ A > μ B .
This is a right tail test.
Calculate the degrees of freedom:
The critical value from a t-table or Excel is approximately1.67, so the reject region is > 1.67 (see the appendix at the end of the book for the Excel function TINV).
Calculate the test statistic: t =
Determine whether to reject or not reject the null hypothesis. In this case, the null hypothesis is rejected since the test statistic is in the reject region.
State the conclusion: the data indicate that the mean response for Method A is larger than the mean response for Method B at the 0.05 significance level.
That α = 0.05 means that 5% of the times that this set of data occurs, it is incorrect to reject the null hypothesis.

CAVEAT 1.3
The hypothesis test does not prove that the mean response for Method A is larger than the mean response for Method B.
Significance Levels
A hypothesis test does not always provide the correct decision. Two types of errors can occur:
• Type I: Rejection of a true null hypothesis
• Type II: Failure to reject a false null hypothesis
The probability that a hypothesis test results in a Type I error is denoted by α and is called the significance level. The probability that a hypothesis test results in a Type II error is denoted by β .
It is possible to decrease both α and β by using a larger sample size.

CAVEAT 1.4
When using a hypothesis test to analyze a particular data set, specifying a smaller significance level, α , increases the probability of a Type II error, β .
FOR EXAMPLE:
Example 1.3
A quality improvement team is tasked with reducing the percentage of units that are rejected due to surface scratches. The current reject rate is 15%. One possible cause of scratches is the design of a holding fixture. The prototype for a new holding fixture design is used to produce a sample of 1000 items. The sample is inspected and 137 units are rejected for scratches. The team uses a hypothesis test to determine whether there has been a reduction in the rejection rate at the 0.10 significance level.
1.2 HYPOTHESIS TEST FOR A POPULATION PROPORTION
Use the following notation:
n = The sample size = 1000
p 0 = old defect rate = 0.15
= sample defect rate = 0.137
p = the new population reject rate
Verify that the conditions for using the test have been met.
n( p 0 ) > 5 and n(1 − p 0 ) > 5 are both true in this example.
State the null hypothesis. In this case, it is Ho: p = 0.15. The alternative hypothesis is Ha: p < 0.15.
α = 0.10.
Critical value is Z 0.10 = − 1.28 for the left tail test. 2 The null hypothesis can be rejected if the test statistic is < − 1.28.
Calculate the test statistic:
Determine whether to reject or not reject the null hypothesis. Since the test statistic is not in the reject region, the null hypothesis can’t be rejected at the 0.10 significance level.
State the conclusion: the data do not support the rejection of the null hypothesis that the defect level is unchanged at the 0.10 significance level.
The hypothesis test implies that the team can’t conclude that the new fixture design reduced the number of scratches at the 0.10 significance level. Therefore, the team decides not to commit the resources required to produce a new fixture. However, looking at the units that have been rejected for scratches over the past year, they develop five categories of scratches. The categories and the approximate percentage of rejected units with those scratches are as follows:
Tapered scratches 24%
Vertical scratches 27%
Diagonal scratches 23%
Parallel pair scratches 17%
J-shaped scratches 9%
(No units had more t