Solution Manual for Fundamentals of Biostatistics, 8th Edition

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Complete Solutions Manualto AccompanyFundamentals of BiostatisticsEIGHTH EDITIONBernard RosnerHarvard University,Cambridge, MAPrepared byRoland A. MatsouakaDuke University, Durham, NC

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ContentsChapter 2 Descriptive Statistics ....................................................................................................... 2Chapter 3 Probability ..................................................................................................................... 21Chapter 4 Discrete Probability Distributions ................................................................................. 43Chapter 5 Continuous Probability Distributions............................................................................ 65Chapter 6 Estimation...................................................................................................................... 93Chapter 7 Hypothesis Testing: One-Sample Inference............................................................... 119Chapter 8 Hypothesis Testing: Two-Sample Inference .............................................................. 146Chapter 9 Nonparametric Methods .............................................................................................. 192Chapter 10 Hypothesis Testing: Categorical Data ....................................................................... 216Chapter 11 Regression and Correlation Methods ........................................................................ 267Chapter 12 Multisample Inference............................................................................................... 322Chapter 13 Design and Analysis Techniques for Epidemiologic Studies.................................... 358Chapter 14 Hypothesis Testing: Person-Time Data.................................................................... 413

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2DESCRIPTIVESTATISTICS2.1We havexxni215258 6.daysmediann12thlargest observation = 13th largest observation = 8 days2.2We have thats2xix2i1252458.6248.62247842432.67sstandard deviation =variancedays5 72.rangelargestsmallest observationdays303272.3Suppose we divide the patients according to whether or not they received antibiotics, and calculate themean and standard deviation for each of the two subsamples:xsnAntibiotics11.578.817No antibiotics7.443.7018Antibiotics -x78.503.736It appears that antibiotic users stay longer in the hospital. Note that when we remove observation 7, thetwo standard deviations are in substantial agreement, and the difference in the means is not thatimpressive anymore. This example shows thatxands2are not robust; that is, their values are easilyaffected by outliers, particularly in small samples. Therefore, we would not conclude that hospital stay isdifferent for antibiotic users vs. non-antibiotic users.

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CHAPTER 2/DESCRIPTIVE STATISTICS32.4-2.7Changing the scale by a factorcwill multiply each data valuexibyc, changing it tocxi. Again the sameindividual’s value will be at the median and the same individual’s value will be at the mode, but thesevalues will be multiplied byc. The geometric mean will be multiplied bycalso, as can easily be shown:Geometric meanold geometric mean[()()()]()()///cxcxcxc xxxc xxxcnnnnnnn121121121The range will also be multiplied byc.For example, ifc2we have:Original Scale–3 –2 –10123xiScale2–6–4–20264xi2.8We first read the data file “running time” in R> require(xlsx)> running<-na.omit(read.xlsx("C:/Data_sets/running_time.xlsx",1,header=TRUE))Let us print the first observations> head(running)weektime11 12.8022 12.2033 12.2544 12.1855 11.5366 12.47The mean 1-mile running time over 18 weeks is equal to 12.09 minutes:> mean(running$time)[1] 12.088892.9The standard deviation is given by> sd(running$time)[1] 0.38741812.10Let us first create the variable “time_100” and then calculate its mean and standard deviation> running$time_100=100*running$time> mean(running$time_100)[1] 1208.889> sd(running$time_100)[1] 38.741812.11Let us to construct the stem-and-leaf plot in R using the stem.leaf command from the package “aplpack”> require(aplpack)

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CHAPTER 2/DESCRIPTIVE STATISTICS4> stem.leaf(running$time_100, unit=1, trim.outliers=FALSE)1 | 2: represents 12leaf unit: 1n: 182115 | 373116 | 75117 | 237118 | 038119 | 2(1)120 | 89121 | 88122 | 056123 | 034124 | 73125 | 52126 | 7127 |1128 | 0Note: one can also use the standard command stem (which does require the “aplpack” package) to get a similar plot> stem(running$time_100, scale = 4)2.12The quantiles of the running times are> quantile(running$time)0%25%50%75%100%11.5300 11.7475 12.1300 12.3225 12.8000An outlying value is identify has any value x such thatxupper quartile+1.5(upper quartile-lower quartile)12.321.5(12.3211.75)12.320.8513.17Since 12.97 minutes is smaller than the largest nonoutlying value(13.17 minutes), this running time recorded in his first week ofrunning in the spring is not an outlying value relative to thedistribution of running times recorded the previous year.2.13The mean isxxi244692419 54.mg dL2.14We have thats2(xixi124)223(4919.54)2(1219.54)2326495.9623282.43s282.4316.81 mg/dL2.15We provide two rows for each stem corresponding to leaves 5-9 and 0-4 respectively. We have11.611.812.012.212.412.612.8Box plot of running timesTime

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CHAPTER 2/DESCRIPTIVE STATISTICS5Stem-and-leaf plotCumulativefrequency+49824+4122+36521+32119+27817+21315+1969913+13329+0886+024008310322.16We wish to compute the average of the (24/2)th and (24/2 + 1)th largest valuesaverage of the 12thand 13th largest points. We note from the stem-and-leaf plot that the 13th largest point counting from thebottom is the largest value in the upper1 row19. The 12th largest pointthe next largest value in thisrow19. Thus, the median1919219 mg dL.2.17We first must compute the upper and lower quartiles. Because24 75 10018is an integer, the upperquartile=averageofthe18thand19thlargestvalues3231231 5..Similarly,because24 25 1006 isaninteger,thelowerquartileaverageofthe6thand7thsmallestpoints812210.Second, we identify outlying values. An outlying value is identified as any valuexsuch thatxupper quartileupper quartilelower quartile)1 531 51 531 51031 532 2563 75.(..(.)...orxlower quartileupper quartilelower quartile) 15101 5315101032 2522 25.(.(.)..From the stem-and-leaf plot, we note that the range is from13 to49. Therefore, there are no outlyingvalues. Thus, the box plot is as follows:Stem-and-leaf plotCumulativefrequencyBox plot+49824|+4122|+36521|+32119+27817||+21315||+1969913+13329+0886|+024|0|083|1032|

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CHAPTER 2/DESCRIPTIVE STATISTICS6Comments:The distribution is reasonably symmetric, since the mean19 5419.mg dLmg dLmedian.Thisisalsomanifestedbythepercentilesofthedistributionsincetheupperquartilemedian31 51912 5..medianlower quartile19109. The box plot looks deceptivelyasymmetric, since 19 is the highest value in the upper1 row and 10 is the lowest value in the lower1row.2.18To compute the median cholesterol level, we construct a stem-and-leaf plot of the before-cholesterolmeasurements as follows.Stem-and-leaf plotCumulativefrequency25024244232368222242202120518195277171801317812166988711115981514521371Based on the cumulative frequency column, we see that the medianaverage of the 12th and 13th largestvalues1781802179mg/dL. Therefore, we look at the change scores among persons with baselinecholesterol179 mg/dL and < 179 mg/dL, respectively. A stem-and-leaf plot of the change scores inthese two groups is given as follows:Baseline179 mg/dLBaseline< 179 mg/dLStem-and-leaf plotStem-and-leaf plot+498+4+4+41+365+3+32+31+278+2+21+23+1699+19+1+1332+08+08+0+02000081103Clearly, from the plot, the effect of diet on cholesterol is much greater among individuals who start withrelatively high cholesterol levels (179 mg/dL) versus those who start with relatively low levels(< 179 mg/dL). This is also evidenced by the mean change in cholesterol levels in the two groups, whichis 28.2 mg/dL in the179 mg/dL group and 10.9 mg/dL in the < 179 mg/dL group. We will bediscussing the formal statistical methods for comparing mean changes in two groups in our work on two-sample inference in Chapter 8.

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CHAPTER 2/DESCRIPTIVE STATISTICS72.19We first calculate the difference scores between the two positions:SubjectnumberSubjectSystolicdifferencescoreDiastolicdifferencescore1B.R.A.682J.A.B.+223F.L.B.+6+44V.P.B.+845M.F.B.+8+26E.H.B.+12+47G.C.+1008M.M.C.029T.J.F.2810R.R.F.+4211C.R.F.+8212E.W.G.+14+413T.F.H.+21414E.J.H.+6215H.B.H.+26016R.T.K.+8+817W.E.L.+10+418R.L.L.+12+219H.S.M.+14+820V.J.M.8221R.H.P.+10+1422R.C.R.+14+423J.A.R.+14024A.K.R.+4+425T.H.S.+6+426O.E.S.+16+227R.E.S.+28+1628E.C.T.+18429J.H.T.+14+430F.P.V.+4631P.F.W.+12+632W.J.W.+84Second, we calculate the mean difference scores:xsysmm Hg 6832282328 8.xdiasmm Hg  843230320 9.The median difference scores are given by the average of the 16th and 17th largest values. Thus,medianmm Hgsys8828medianmm Hgdias0221

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CHAPTER 2/DESCRIPTIVE STATISTICS82.20The stem-and-leaf and box plots allowing two rows for each stem are given as follows:Systolic Blood PressureStem-and-leaf plotCumulativefrequencyBox plot26832|2|16830|120402404442280688868681702042449023|0682|Medianupper quartilelower quartile814142144424,,,outlying values:xx 14151442941514411.().()or. Since the range of values is from –8 to +28,there are no outlying values for systolic blood pressure.Diastolic Blood PressureStem-and-leaf plotCumulativefrequencyBox plot1632014310088630|04240404240442427+02422222441308864|1410Medianupper quartilelower quartile  144242222,,,outlying values:xx  41 54213 021 54211 0.()..().or. The values +16, +14 and –14 are outlyingvalues.2.21Systolic blood pressure clearly seems to be higher in the supine (recumbent) position than in the standingposition. Diastolic blood pressure appears to be comparable in the two positions. The distributions areeach reasonably symmetric.2.22The upper and lower deciles for postural change in systolic blood pressure (SBP) are 14 and 0. Thus, thenormal range for postural change in SBP is014x. The upper and lower deciles for postural changein diastolic blood pressure (DBP) are 8 and –6.Thus, the normal range for postural change in DBP is68x.2.23IdAgeFEVHgtSexSmoke30191.708570045181.72467.500......61951152.278600163241164.504721071141175.638701071142164.872721173041164.27671173042153.727681173751182.8536000

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CHAPTER 2/DESCRIPTIVE STATISTICS975852162.795630177151153.21166.500MEAN9.9311932.6367861.143580.5137610.099388MEDIAN102.547561.5SD2.9539350.8670595.7035131815129639080706050403020100AgeFrequencyHistogram of Age654321FEVBoxplot of FEV75706560555045HgtBoxplot of Hgt10350300250200150100500SexCountChart of Sex106005004003002001000SmokeCountChart of Smoke2.24Results for Sex = 0VariableAgeMeanStDevMinimumMedianMaximumFEV31.0720*1.07201.07201.072041.3160.2900.8391.4041.57751.35990.25130.79101.37151.704061.64770.21821.33801.67202.102071.83300.31361.37001.74202.564082.14900.40461.29202.19002.993092.37530.44071.59102.38103.2230102.68140.43041.45802.68953.4130112.84820.42932.08102.82203.7740122.94810.36792.34702.88903.8350133.06560.43212.21603.11353.8160142.9620.3832.2362.9973.428152.7610.4152.1982.7833.330163.0580.3972.6082.9423.674173.5000*3.50003.50003.5000182.94700.11992.85302.90603.0820193.43200.12303.34503.43203.5190Results for Sex = 1VariableAgeMeanStDevMinimumMedianMaximumFEV31.4040*1.40401.40401.404041.1960.5240.7961.0041.78951.74470.23361.35901.79202.115061.66500.23041.33801.65802.2620

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CHAPTER 2/DESCRIPTIVE STATISTICS1071.91170.35941.16501.90502.578082.07560.37671.42902.06902.927092.48220.50861.55802.45703.8420102.69650.60201.66502.60804.5910113.23040.64591.69403.20604.6370123.5090.8711.9163.5305.224134.0110.6902.5314.0455.083143.9310.6352.2763.8824.842154.2890.6443.7274.2795.793164.1930.4373.6454.2704.872174.4101.0063.0824.4295.638184.23670.15974.08604.22004.4040195.1020*5.10205.10205.1020-----------------------------------------------------------------------------------------------------------------------------Results for Sex = 0VariableHgtMeanFEV46.01.072046.51.196048.01.11049.01.419350.01.337851.01.580051.51.47452.01.38952.51.57753.01.688753.51.415054.01.640854.51.748355.01.631355.52.03656.01.65156.51.787557.01.903757.51.930058.02.193458.51.944059.02.199659.52.51760.02.565960.52.556361.02.698161.52.62662.02.786162.52.777763.02.726663.52.99564.02.973164.52.86465.03.09065.42.434065.53.15466.02.98466.53.284367.03.16767.52.92268.03.21468.53.330069.53.835071.02.538020151052015105642706050642706050Age, 0FEVAge, 1Hgt, 0Hgt, 1Scatterplot of FEV vs Age, HgtPanel variable: Sex

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CHAPTER 2/DESCRIPTIVE STATISTICS11Results for Sex = 1VariableHgtMeanFEV47.00.98148.01.27049.51.425050.01.79450.51.53651.01.68351.51.51452.01.591552.51.710053.01.664653.51.97454.01.780954.51.838055.01.803455.51.807056.02.02556.51.87957.02.087557.51.82958.02.016958.52.13159.02.35059.52.51560.02.27960.52.325361.02.469961.52.541062.02.65862.52.82963.02.87763.52.75764.02.69764.53.10065.02.77065.53.034366.03.11566.53.35367.03.77967.53.61268.03.87868.53.87269.04.02269.53.74370.04.19770.53.93171.04.31071.54.720072.04.36172.54.272073.05.25573.53.645074.04.654---------------------------------------------------------------------------------------------------------------------------------Descriptive Statistics: FEVResults for Sex = 0

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CHAPTER 2/DESCRIPTIVE STATISTICS12VariableSmokeMeanStDevFEV02.37920.639312.96590.4229Results for Sex = 1VariableSmokeMeanStDevFEV02.73440.974113.7430.8892.25Looking at the scatterplot of FEV vs. Age, we find that FEV increases with age for both boys and girls, atapproximately the same rate. However, the spread (standard deviation) of FEV values appears to behigher in male group than in the female group.2.26VariableMeanStDevMedianSat. Fat - DR14.5577.53612.000Sat. Fat - FFQ7.8989.6953.159Tot. Fat - DR64.2389.89463.500Tot. Fat - FFQ15.2127.001.00Alcohol - DR2.4706.3140.000Alcohol - FFQ8.95112.2554.550Calories - DR1619.9323.41606.0Calories - FFQ1371.7482.11297.6Alcohol - FFQAlcohol - DRTot. Fat - FFQTot. Fat - DRSat. Fat - FFQSat. Fat - DR140120100806040200DataBoxplot of Sat. Fat, Tot. Fat, and Alcohol2.27SexSmoke101010654321FEV01SexBoxplot of FEVCalories - FFQCalories - DR30002500200015001000500DataBoxplot of Calories

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CHAPTER 2/DESCRIPTIVE STATISTICS1360453015050403020101501005001201008060406045301504836241203000200010002500200015001000Sat. Fat - DR*Sat. Fat - FFQTot. Fat - DR*Tot. Fat - FFQAlcohol - DR*Alcohol - FFQCalories - DR*Calories - FFQScatterplot of DR vs. FFQ valuesIf FFQ were a perfect substitute for DR, the points would line up in a straight line. If the two wereunrelated, then we would expect to see a random pattern in each panel.The scatterplots shown above seemto suggest that the DR and FFQ values are not highly related.2.28The 5x5 tables below show the number of people classified into a particular combination of quintilecategories. For each table, the rows represent the quintiles of the DR, and the columns represent quintiles ofthe FFQ. Overall, we get the same impression that there is weak concordance between the two measures.However, we do notice that the agreement is greatest for the two measures with regards to alcoholconsumption. Also, we note the relatively high level of agreement at the extremes of each nutrient; forexample, the (1,1) and (5,5) cells generally contain the highest values.Tabulated statistics: SFDQuin, SFFQuinRows: SFDQuinColumns: SFFQuin12345All11589213521066853534789634461069435503671834All3534353534173Cell Contents:CountTabulated statistics: TFDQuin, TFFQuinRows: TFDQuinColumns: TFFQuin12345All11398513629571033434108663448639935515851534

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CHAPTER 2/DESCRIPTIVE STATISTICS14All3535343534173Cell Contents:CountTabulated statistics: AlcDQuin, AlcFQuinRows: AlcDQuinColumns: AlcFQuin12345All12852003526236003530914101344011016835500082634All3438323435173Cell Contents:CountTabulated statistics: CalDQuin, CalFQuinRows: CalDQuinColumns: CalFQuin12345All1101184235211497435359686344487610355534101234All35353435341732.29Descriptive Statistics: Total Fat Density DR, Total Fat Density FFQVariableMeanStDevMedianTotal Fat Density DR38.0664.20538.646Total Fat Density FFQ36.8556.72936.366605040302010050403020100Total Fat Density FFQTotal Fat Density DR00Scatterplot of Total Fat Density DR vs Total Fat Density FFQ

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