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168 CHAPTER 6 ESTIMATION Precision The quantity obtained by multiplying the reliability factor by the standard error of the mean is called the precision of the estimate. This quantity is also called the margin of error. EXAMPLE 6.2.2 A physical therapist wished to estimate, with 99 percent confidence, the mean maximal strength of a particular muscle in a certain group of individuals. He is willing to assume that strength scores are approximately normally distributed with a variance of 144. A sample of 15 subjects who participated in the experiment yielded a mean of 84.3. Solution: The z value corresponding to a confidence coefficient of .99 is found in Appendix Table D to p be 2.58. This is our reliability coefficient. The standard error is s x ¼ 12= ffiffiffiffiffi 15 ¼ 3:0984. Our 99 percent confidence interval for m, then, is 84:3 2:58ð3:0984Þ 84:3 8:0 76:3; 92:3 We say we are 99 percent confident that the population mean is between 76.3 and 92.3 since, in repeated sampling, 99 percent of all intervals that could be constructed in the manner just described would include the population mean. & Situations in which the variable of interest is approximately normally distributed with a known variance are quite rare. The purpose of the preceding examples, which assumed that these ideal conditions existed, was to establish the theoretical background for constructing confidence intervals for population means. In most practical situations either the variables are not approximately normally distributed or the population variances are not known or both. Example 6.2.3 and Section 6.3 explain the procedures that are available for use in the less than ideal, but more common, situations. Sampling from Nonnormal Populations As noted, it will not always be possible or prudent to assume that the population of interest is normally distributed. Thanks to the central limit theorem, this will not deter us if we are able to select a large enough sample. We have learned that for large samples, the sampling distribution of x is approximately normally distributed regardless of how the parent population is distributed. EXAMPLE 6.2.3 Punctuality of patients in keeping appointments is of interest to a research team. In a study of patient flow through the offices of general practitioners, it was found that a sample of 35 patients was 17.2 minutes late for appointments, on the average. Previous research had shown the standard deviation to be about 8 minutes. The population distribution was felt to be nonnormal. What is the 90 percent confidence interval for m, the true mean amount of time late for appointments?
6.2 CONFIDENCE INTERVAL FOR A POPULATION MEAN 169 Solution: Since the sample size is fairly large (greater than 30), and since the population standard deviation is known, we draw on the central limit theorem and assume the sampling distribution of x to be approximately normally distributed. From Appendix Table D we find the reliability coefficient corresponding to a confidence coefficient p of .90 to be about 1.645, if we interpolate. The standard error is s x ¼ 8= ffiffiffiffiffi 35 ¼ 1:3522, so that our 90 percent confidence interval for m is 17:2 1:645ð1:3522Þ 17:2 2:2 15:0; 19:4 & Frequently, when the sample is large enough for the application of the central limit theorem, the population variance is unknown. In that case we use the sample variance as a replacement for the unknown population variance in the formula for constructing a confidence interval for the population mean. Computer Analysis When confidence intervals are desired, a great deal of time can be saved if one uses a computer, which can be programmed to construct intervals from raw data. EXAMPLE 6.2.4 The following are the activity values (micromoles per minute per gram of tissue) of a certain enzyme measured in normal gastric tissue of 35 patients with gastric carcinoma. .360 1.189 .614 .788 .273 2.464 .571 1.827 .537 .374 .449 .262 .448 .971 .372 .898 .411 .348 1.925 .550 .622 .610 .319 .406 .413 .767 .385 .674 .521 .603 .533 .662 1.177 .307 1.499 We wish to use the MINITAB computer software package to construct a 95 percent confidence interval for the population mean. Suppose we know that the population variance is .36. It is not necessary to assume that the sampled population of values is normally distributed since the sample size is sufficiently large for application of the central limit theorem. Solution: We enter the data into Column 1 and proceed as shown in Figure 6.2.2 . These instructions tell the computer that the reliability factor is z, that a 95 percent confidence interval is desired, that the population standard deviation is .6, and that the data are in Column 1. The output tells us that the sample mean is .718, the p sample standard deviation is .511, and the standard error of the mean, s= ffiffi pffiffiffiffiffi n is :6= 35 ¼ :101. & We are 95 percent confident that the population mean is somewhere between .519 and .917. Confidence intervals may be obtained through the use of many other software packages. Users of SAS ® , for example, may wish to use the output from PROC MEANS or PROC UNIVARIATE to construct confidence intervals.
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TENTH EDITION BIOSTATISTICS A Found
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VP & EXECUTIVE PUBLISHER: ACQUISITI
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PREFACE This 10th edition of Biosta
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PREFACE ix SUPPLEMENTS Instructor
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BRIEF CONTENTS 1 INTRODUCTION TO BI
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xiv CONTENTS 6.3 The t Distribution
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CHAPTER1 INTRODUCTION TO BIOSTATIST
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1.2 SOME BASIC CONCEPTS 3 Sources o
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1.3 MEASUREMENT AND MEASUREMENT SCA
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1.4 SAMPLING AND STATISTICAL INFERE
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1.5 THE SCIENTIFIC METHOD AND THE D
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1.6 COMPUTERS AND BIOSTATISTICAL AN
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REVIEW QUESTIONS AND EXERCISES 17 a
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CHAPTER2 DESCRIPTIVE STATISTICS CHA
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2.2 THE ORDERED ARRAY 21 TABLE 2.2.
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2.3 GROUPED DATA: THE FREQUENCY DIS
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EXERCISES 31 3 ¼ an abundance of s
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EXERCISES 33 25.21 30.49 27.38 36.4
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EXERCISES 35 2.3.7 In a study of ph
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EXERCISES 37 0.6870 0.6900 0.6910 0
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2.4 DESCRIPTIVE STATISTICS: MEASURE
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EXERCISES 53 2.5.1 Porcellini et al
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SUMMARY OF FORMULAS FOR CHAPTER 2 5
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REVIEW QUESTIONS AND EXERCISES 57 R
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REVIEW QUESTIONS AND EXERCISES 59 1
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REFERENCES 63 DRINK TOUNCES TGRAMS
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CHAPTER3 SOME BASIC PROBABILITY CON
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3.2 TWO VIEWS OF PROBABILITY: OBJEC
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3.4 CALCULATING THE PROBABILITY OF
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EXERCISES 77 (d) If we pick a subje
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3.5 BAYES’ THEOREM, SCREENING TES
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3.5 BAYES’ THEOREM, SCREENING TES
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EXERCISES 83 EXERCISES 3.5.1 A medi
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REVIEW QUESTIONS AND EXERCISES 87 (
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REVIEW QUESTIONS AND EXERCISES 89 i
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REFERENCES 91 Applications Referenc
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4.2 PROBABILITY DISTRIBUTIONS OF DI
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4.3 THE BINOMIAL DISTRIBUTION 99 TA
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4.3 THE BINOMIAL DISTRIBUTION 101 E
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4.3 THE BINOMIAL DISTRIBUTION 103 f
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EXAMPLE 4.3.4 According to a June 2
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EXERCISES 107 Data: C1: 0 1 2 3 4 5
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4.4 THE POISSON DISTRIBUTION 109 pr
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4.4 THE POISSON DISTRIBUTION 111 or
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4.5 CONTINUOUS PROBABILITY DISTRIBU
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4.5 CONTINUOUS PROBABILITY DISTRIBU
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7.1 INTRODUCTION 219 DEFINITION The
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7.1 INTRODUCTION 221 Evaluate data
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7.2 HYPOTHESIS TESTING: A SINGLE PO
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EXERCISES 233 Dialog box: Session c
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EXERCISES 235 7.2.10 Following a we
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7.3 HYPOTHESIS TESTING: THE DIFFERE
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EXAMPLE 7.3.2 The purpose of a stud
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EXERCISES 245 The SAS System The TT
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EXERCISES 247 Primary Hypertensive
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7.4 PAIRED COMPARISONS 249 7.3.10 R
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7.4 PAIRED COMPARISONS 251 where d
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7.4 PAIRED COMPARISONS 253 a = .05
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EXERCISES 255 7.4.1 Ellen Davis Jon
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EXERCISES 263 MINITAB Output Test a
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7.8 HYPOTHESIS TESTING: THE RATIO O
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7.8 HYPOTHESIS TESTING: THE RATIO O
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EXERCISES 271 EXERCISES In the foll
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7.9 THE TYPE II ERROR AND THE POWER
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7.9 THE TYPE II ERROR AND THE POWER
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7.10 DETERMINING SAMPLE SIZE TO CON
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EXERCISES 279 We set the right-hand
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REVIEW QUESTIONS AND EXERCISES 283
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REFERENCES 301 A-17. JOHN S. MORLEY
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REFERENCES 303 A-56. LUIGI BENINI,
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8.1 INTRODUCTION 305 8.1 INTRODUCTI
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8.1 INTRODUCTION 307 who received d
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8.2 THE COMPLETELY RANDOMIZED DESIG
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Dependent Variable: Selenium Tukey
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EXERCISES 329 Elbow Angle (Degrees)
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EXERCISES 331 Age Groups (Years) Ag
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EXERCISES 333 Group 1 Group 2 Group
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Objective The objective in using th
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8.3 THE RANDOMIZED COMPLETE BLOCK D
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EXERCISES 343 The SAS System Analys
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EXERCISES 345 group were randomly a
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8.4 THE REPEATED MEASURES DESIGN 34
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EXERCISES 357 Is there sufficient e
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8.5 THE FACTORIAL EXPERIMENT 359 In
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8.5 THE FACTORIAL EXPERIMENT 361 In
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8.5 THE FACTORIAL EXPERIMENT 363 Be
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8.5 THE FACTORIAL EXPERIMENT 365 TA
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8.5 THE FACTORIAL EXPERIMENT 367 Ro
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EXERCISES 369 The SAS System Analys
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EXERCISES 371 Physical Psychiatric
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8.6 SUMMARY 373 Percent Change in C
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REFERENCES 409 Applications Referen
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REFERENCES 411 Postmenopausal Women
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CHAPTER9 SIMPLE LINEAR REGRESSION A
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9.2 THE REGRESSION MODEL 415 some e
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9.3 THE SAMPLE REGRESSION EQUATION
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EXERCISES EXERCISES 423 The Least-S
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EXERCISES 425 CoaguChek Hospital Co
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9.4 EVALUATING THE REGRESSION EQUAT
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9.5 USING THE REGRESSION EQUATION I
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9.5 USING THE REGRESSION EQUATION 4
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9.6 THE CORRELATION MODEL 445 9.6 T
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9.7 THE CORRELATION COEFFICIENT 447
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EXERCISES 455 is employed. We first
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EXERCISES 457 9.7.3 In the study by
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9.8 SOME PRECAUTIONS 459 X Y X Y 5.
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9.9 SUMMARY 461 variance, which tes
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9.7.3 t statistic for correlation c
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REFERENCES 487 A-4. ROBERT B. PARKE
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CHAPTER10 MULTIPLE REGRESSION AND C
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Assumptions follows. 10.2 THE MULTI
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10.3 OBTAINING THE MULTIPLE REGRESS
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10.3 OBTAINING THE MULTIPLE REGRESS
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EXERCISES 497 EXERCISES Obtain the
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EXERCISES 499 10.3.3 In a study of
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10.4 EVALUATING THE MULTIPLE REGRES
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10.5 USING THE MULTIPLE REGRESSION
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EXERCISES 509 and has an education
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10.6 THE MULTIPLE CORRELATION MODEL
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This equation may be used for estim
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EXERCISES 521 HIV DNA Blood (Y) HIV
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SUMMARY OF FORMULAS FOR CHAPTER 10
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10.6.7 t statistic for testing hypo
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CHAPTER11 REGRESSION ANALYSIS: SOME
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11.1 INTRODUCTION 541 TABLE 11.1.1
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11.2 QUALITATIVE INDEPENDENT VARIAB
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EXERCISES 557 11.2.1 For subjects u
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11.3 VARIABLE SELECTION PROCEDURES
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11.3 VARIABLE SELECTION PROCEDURES
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EXERCISES 565 CGAGE CGINCOME CGDUR
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EXERCISES 567 11.3.2 Machiel Naeije
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11.4 LOGISTIC REGRESSION 569 REACTI
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11.4 LOGISTIC REGRESSION 571 betwee
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11.4 LOGISTIC REGRESSION 573 The LO
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11.4 LOGISTIC REGRESSION 575 Standa
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11.4 LOGISTIC REGRESSION 577 Parame
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11.4 LOGISTIC REGRESSION 579 Model
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EXERCISES EXERCISES 581 Further Rea
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REFERENCES 597 KNOW ¼ Knowledge. S
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12.1 INTRODUCTION 12.2 THE MATHEMAT
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12.2 THE MATHEMATICAL PROPERTIES OF
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12.3 TESTS OF GOODNESS-OF-FIT 605 W
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12.3 TESTS OF GOODNESS-OF-FIT 607 x
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12.3 TESTS OF GOODNESS-OF-FIT 609 9
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that the first expected frequency i
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12.3 TESTS OF GOODNESS-OF-FIT 613 T
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12.3 TESTS OF GOODNESS-OF-FIT 615 V
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EXERCISES 617 EXERCISES 12.3.1 The
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12.4 TESTS OF INDEPENDENCE 619 12.3
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Observed Versus Expected Frequencie
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12.4 TESTS OF INDEPENDENCE 623 Comp
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12.4 TESTS OF INDEPENDENCE 625 Note
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12.4 TESTS OF INDEPENDENCE 627 9. C
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EXERCISES 629 Performed Baseline Te
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12.5 TESTS OF HOMOGENEITY 631 TABLE
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12.5 TESTS OF HOMOGENEITY 633 8. St
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EXERCISES 635 12.5.2 Coughlin et al
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Hypotheses alternatives. 12.6 THE F
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12.6 THE FISHER EXACT TEST 639 TABL
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EXERCISES 653 Smoking_status * Obse
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12.8 SUMMARY 655 collected data on
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REFERENCES 667 12. R. LATSCHA, “T
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REFERENCES 669 A-35. CRAIG R. COHEN
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13.1 INTRODUCTION 671 2. be able to
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13.3 THE SIGN TEST 673 13.3 THE SIG
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13.3 THE SIGN TEST 675 TABLE 13.3.2
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13.3 THE SIGN TEST 677 EXAMPLE 13.3
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13.3 THE SIGN TEST 679 one sample i
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13.4 THE WILCOXON SIGNED-RANK TEST
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13.4 THE WILCOXON SIGNED-RANK TEST
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13.5 THE MEDIAN TEST 687 TABLE 13.5
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EXAMPLE 13.6.1 A researcher designe
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13.6 THE MANN-WHITNEY TEST 693 to t
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13.6 THE MANN-WHITNEY TEST 695 Dial
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EXERCISES 697 deficit in Sprague-Da
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13.7 THE KOLMOGOROV-SMIRNOV GOODNES
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EXERCISES 703 1.0 .9 Cumulative rel
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13.8 THE KRUSKAL-WALLIS ONE-WAY ANA
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13.8 THE KRUSKAL-WALLIS ONE-WAY ANA
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EXERCISES 709 Data: C1: 12.22 28.44
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EXERCISES 711 Improvement (Control)
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13.9 THE FRIEDMAN TWO-WAY ANALYSIS
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EXERCISES 717 Subject Area Student
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13.10 THE SPEARMAN RANK CORRELATION
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EXERCISES 725 Rank by Rank by Area
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13.11 NONPARAMETRIC REGRESSION ANAL
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REFERENCES 747 cardiac operations.
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REFERENCES 749 A-14. WENDY GANTT an
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14.2 TIME-TO-EVENT DATA AND CENSORI
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14.3 THE KAPLAN-MEIER PROCEDURE 757
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14.3 THE KAPLAN-MEIER PROCEDURE 759
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EXERCISES 761 1.0 .8 73% Low-grade
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14.4 COMPARING SURVIVAL CURVES 763
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14.4 COMPARING SURVIVAL CURVES 765
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EXERCISES 767 EXERCISES 14.4.1 If a
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14.5 COX REGRESSION: THE PROPORTION
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REFERENCES 777 REFERENCES Methodolo
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A-2 APPENDIX STATISTICAL TABLES
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ANSWERS TO ODD-NUMBERED EXERCISES A
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I-2 INDEX Confidence coefficient, 1
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I-4 INDEX Median test, 686-689 MINI
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I-6 INDEX SAS: and chi-square analy
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Biostatistics

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