Fuzzy Statistics

1. 1 Introduction This book is written in four major divisions. The first part is the introductory chapters consisting of Chapters 1 and 2. In part two, Chapters 3-11, we develop fuzzy estimation. For example, in Chapter 3 we construct a fuzzy estimator for the mean of a normal distribution assuming the variance is known. More details on fuzzy estimation are in Chapter 3 and then after Chapter 3, Chapters 4-11 can be read independently. Part three, Chapters 12- 20, are on fuzzy hypothesis testing. For example, in Chapter 12 we consider the test Ho : /1 = /10 verses HI : /1 f=- /10 where /1 is the mean of a normal distribution with known variance, but we use a fuzzy number (from Chapter 3) estimator of /1 in the test statistic. More details on fuzzy hypothesis testing are in Chapter 12 and then after Chapter 12 Chapters 13-20 may be read independently. Part four, Chapters 21-27, are on fuzzy regression and fuzzy prediction. We start with fuzzy correlation in Chapter 21. Simple linear regression is the topic in Chapters 22-24 and Chapters 25-27 concentrate on multiple linear regression. Part two (fuzzy estimation) is used in Chapters 22 and 25; and part 3 (fuzzy hypothesis testing) is employed in Chapters 24 and 27. Fuzzy prediction is contained in Chapters 23 and 26. A most important part of our models in fuzzy statistics is that we always start with a random sample producing crisp (non-fuzzy) data.

Fuzzy Sets.- Estimate ?, Variance Known.- Estimate ?, Variance Unknown.- Estimate p, Binomial Population.- Estimate ?2 from a Normal Population.- Estimate µ 1 — µ 2, Variances Known.- Estimate ? 1 — ? 2, Variances Unknown.- Estimate d =? 1 — ? 2, Matched Pairs.- Estimate p 1 — p 2, Binomial Populations.- Estimate ? 1 2 /? 2 2 , Normal Populations.- Tests on µ, Variance Known.- Tests on µ, Variance Unknown.- Tests on p for a Binomial Population.- Tests on ? 2, Normal Population.- Tests ? 1 verses ? 2, Variances Known.- Test ? 1 verses ? 2, Variances Unknown.- Test p 1 = p 2, Binomial Populations.- Test d = µ 1 — µ 2 , Matched Pairs.- Test ? 1 2 verses ? 2 2 , Normal Populations.- Fuzzy Correlation.- Estimation in Simple Linear Regression.- Fuzzy Prediction in Linear Regression.- Hypothesis Testing in Regression.- Estimation in Multiple Regression.- Fuzzy Prediction in Regression.- Hypothesis Testing in Regression.- Summary and Questions.- Maple Commands.