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15-58. For the large-sample approximation to the Wilcoxon signed-rank test, derive the mean and standard deviation of the test statistic used in the procedure. 15-59. Testing for Trends. A turbocharger wheel is manufactured using an investment-casting process. The shaft ts into the wheel opening, and this wheel opening is a critical dimension. As wheel wax patterns are formed, the hard tool producing the wax patterns wears. This may cause growth in the wheel-opening dimension. Ten wheel-opening measurements, in time order of production, are 4.00 (millimeters), 4.02, 4.03, 4.01, 4.00, 4.03, 4.04, 4.02, 4.03, 4.03. (a) Suppose that p is the probability that observation Xi 5 exceeds observation Xi. If there is no upward or downward trend, Xi 5 is no more or less likely to exceed Xi or lie below Xi. What is the value of p (b) Let V be the number of values of i for which Xi 5 Xi. If there is no upward or downward trend in the measurements, what is the probability distribution of V (c) Use the data above and the results of parts (a) and (b) to test H0: there is no trend, versus H1: there is upward trend. Use 0.05. Note that this test is a modi cation of the sign test. It was developed by Cox and Stuart. 15-60. Consider the Wilcoxon signed-rank test, and suppose that n 5. Assume that H0: 0 is true. (a) How many different sequences of signed ranks are possible Enumerate these sequences. (b) How many different values of W are there Find the probability associated with each value of W . (c) Suppose that we de ne the critical region of the test to be to reject H0 if w w* and w* 13. What is the approximate level of this test (d) Does this exercise show how the critical values for the Wilcoxon signed-rank test were developed Explain.
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IMPORTANT TERMS AND CONCEPTS In the E-book, click on any term or concept below to go to that subject. Asymptotic relative ef ciency Kruskal-Wallis test Nonparametric or distribution free procedures Normal approximation to nonparametric tests Paired observations Ranks Rank transformation in ANOVA Sign test Tests on the mean of a symmetric continuous distribution Tests on the median of a continuous distribution Wilcoxon rank-sum test Wilcoxon signed-rank test
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16-1 16-2 16-3 16-4 QUALITY IMPROVEMENT AND STATISTICS STATISTICAL QUALITY CONTROL STATISTICAL PROCESS CONTROL INTRODUCTION TO CONTROL CHARTS 16-4.1 Basic Principles 16-4.2 Design of a Control Chart 16-4.3 Rational Subgroups 16-4.4 Analysis of Patterns on Control Charts 16-5 X AND R OR S CONTROL CHARTS
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Statistical Quality Control
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16-6 16-7 16-8
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CONTROL CHARTS FOR INDIVIDUAL MEASUREMENTS PROCESS CAPABILITY ATTRIBUTE CONTROL CHARTS 16-8.1 P Chart (Control Chart for Proportions) 16-8.2 U Chart (Control Chart for Defects per Unit)
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After careful study of this chapter, you should be able to do the following: 1. Understand the role of statistical tools in quality improvement 2. Understand the different types of variability, rational subgroups, and how a control chart is used to detect assignable causes 3. Understand the general form of a Shewhart control chart and how to apply zone rules (such as the Western Electric rules) and pattern analysis to detect assignable causes 4. Construct and interpret control charts for variables such as X , R, S, and individuals charts 5. Construct and interpret control charts for attributes such as P and U charts 6. Calculate and interpret process capability ratios
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7. Calculate the ARL performance for a Shewhart control chart 8. Construct and interpret a cumulative sum control chart 9. Use other statistical process control problem-solving tools Answers for most odd numbered exercises are at the end of the book. Answers to exercises whose numbers are surrounded by a box can be accessed in the e-text by clicking on the box. Complete worked solutions to certain exercises are also available in the e-text. These are indicated in the Answers to Selected Exercises section by a box around the exercise number. Exercises are also available for some of the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany.
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