R hw
BME 530 Assignment 3 Using R to complete all assignments if applicable. Q1 (3pts). Blood Types. The prevalence of blood types in the US population is O+: 37.4%, A+: 35.7%, B+: 8.5%, AB+: 3.4%, O–: 6.6%, A–: 6.3%, B–: 1.5%, and AB–: 0.6%. 1) A sample of 24 subjects is randomly selected from the US population. What is the probability that 8 subjects are A+. Random variable X describes the number of A+ subjects among 24 selected. Find E(X) and Var (X). 2) Find the probability that among 500 randomly selected subjects, the number of B– subjects is at least 2. 3) Use the Poisson approximation to calculate the probability in 2). Q2 (4pts). 1) Let X be a random variable with the following probability distribution: x -3 6 9 f(x) 1/6 1/2 1/3 Find E(g(x)) and var(g(x)), where g(X) = (2X + 1)2. 2) The density function of a random variable X is ?(?) = { 2(? + 2) 5 , 0 <>< 1 0, ??ℎ?????? find e(x) and var(x). q3 (3pts). leptin (from the greek word leptos, meaning thin) is a 16-kda protein hormone that plays a crucial role in regulating energy intake and energy expenditure, including the regulation (decrease) of appetite and (increase) of metabolism. serum leptin concentrations can be measured in several ways. one approach uses a radioimmunoassay in venous blood samples (linco research inc., st charles, mo). several studies have consistently found women to have higher serum leptin concentrations than do men. for example, among us adults across a broad age range, the mean serum leptin concentration in women is approximately normal ?(12.7??/?, (1.3??/?)2) and in men approximately normal ?(4.6??/?, (0.5??/?)2). 1) what is the probability that leptin concentration in a randomly selected us adult male exceeds 6 ??/?? 2) what proportion of us women have the concentration of leptin in the interval 12.7 ± 2??/?? 3) what interval, symmetric about the mean 12.7??/?, contains leptin concentrations of 95% of adult us women? q4 (5pts). take a sample of size n=10 from the poisson distribution with mean ?=3 and compute the sample mean. repeat this procedure 10,000 times and plot the histogram of sample means. then repeat this procedure for n=100, 1000. is there any trend observed in the histogram of the sample mean as n increases? what is the property of the sampling distribution of the sample mean for n=1000? what are the mean and variance of this distribution? q5 (5pts). use the data in hw1 to make the following statistical inferences about the following attributes in a general heart disease population where the data were generated. 1) construct the 95% confidence intervals for average creatinine phosphokinase (cpk) levels in deceased and alive patients in the population, respectively. 2) construct the 95% confidence intervals for variances of cpk levels in deceased and alive patients in the population, respectively. 3) construct the 95% confidence intervals for proportions of diabetes in deceased and alive patients in the population, respectively. justify the sampling distributions used for the calculation in (1)-(3). you may use the histograms of the cpk levels for 1) and 2). submission. submit a zip file (hw3_name.zip) on blackboard that includes: 1) both r notebook (.rmd) and rendered results (.html). your notebook should include some annotations for the scripts so that a grader knows what the script is for. 2) report document (.pdf) age,anaemia,creatinine_phosphokinase,diabetes,ejection_fraction,high_blood_pressure,platelets,serum_creatinine,serum_sodium,sex,smoking,time,death_event 75,0,582,0,20,1,265000,1.9,130,1,0,4,1 55,0,7861,0,38,0,263358.03,1.1,136,1,0,6,1 65,0,146,0,20,0,162000,1.3,129,1,1,7,1 50,1,111,0,20,0,210000,1.9,137,1,0,7,1 65,1,160,1,20,0,327000,2.7,116,0,0,8,1 90,1,47,0,40,1,204000,2.1,132,1,1,8,1 75,1,246,0,15,0,127000,1.2,137,1,0,10,1 60,1,315,1,60,0,454000,1.1,131,1,1,10,1 65,0,157,0,65,0,263358.03,1.5,138,0,0,10,1 80,1,123,0,35,1,388000,9.4,133,1,1,10,1 75,1,81,0,38,1,368000,4,131,1,1,10,1 62,0,231,0,25,1,253000,0.9,140,1,1,10,1 45,1,981,0,30,0,136000,1.1,137,1,0,11,1 50,1,168,0,38,1,276000,1.1,137,1,0,11,1 49,1,80,0,30,1,427000,1,138,0,0,12,0 82,1,379,0,50,0,47000,1.3,136,1,0,13,1 87,1,149,0,38,0,262000,0.9,140,1,0,14,1 45,0,582,0,14,0,166000,0.8,127,1,0,14,1 70,1,125,0,25,1,237000,1,140,0,0,15,1 48,1,582,1,55,0,87000,1.9,121,0,0,15,1 65,1,52,0,25,1,276000,1.3,137,0,0,16,0 65,1,128,1,30,1,297000,1.6,136,0,0,20,1 68,1,220,0,35,1,289000,0.9,140,1,1,20,1 53,0,63,1,60,0,368000,0.8,135,1,0,22,0 75,0,582,1,30,1,263358.03,1.83,134,0,0,23,1 80,0,148,1,38,0,149000,1.9,144,1,1,23,1 95,1,112,0,40,1,196000,1,138,0,0,24,1 70,0,122,1,45,1,284000,1.3,136,1,1,26,1 58,1,60,0,38,0,153000,5.8,134,1,0,26,1 82,0,70,1,30,0,200000,1.2,132,1,1,26,1 94,0,582,1,38,1,263358.03,1.83,134,1,0,27,1 85,0,23,0,45,0,360000,3,132,1,0,28,1 50,1,249,1,35,1,319000,1,128,0,0,28,1 50,1,159,1,30,0,302000,1.2,138,0,0,29,0 65,0,94,1,50,1,188000,1,140,1,0,29,1 69,0,582,1,35,0,228000,3.5,134,1,0,30,1 90,1,60,1,50,0,226000,1,134,1,0,30,1 82,1,855,1,50,1,321000,1,145,0,0,30,1 60,0,2656,1,30,0,305000,2.3,137,1,0,30,0 60,0,235,1,38,0,329000,3,142,0,0,30,1 70,0,582,0,20,1,263358.03,1.83,134,1,1,31,1 50,0,124,1,30,1,153000,1.2,136,0,1,32,1 70,0,571,1,45,1,185000,1.2,139,1,1,33,1 72,0,127,1,50,1,218000,1,134,1,0,33,0 60,1,588,1,60,0,194000,1.1,142,0,0,33,1 50,0,582,1,38,0,310000,1.9,135,1,1,35,1 51,0,1380,0,25,1,271000,0.9,130,1,0,38,1 60,0,582,1,38,1,451000,0.6,138,1,1,40,1 80,1,553,0,20,1,140000,4.4,133,1,0,41,1 57,1,129,0,30,0,395000,1,140,0,0,42,1 68,1,577,0,25,1,166000,1,138,1,0,43,1 53,1,91,0,20,1,418000,1.4,139,0,0,43,1 60,0,3964,1,62,0,263358.03,6.8,146,0,0,43,1 70,1,69,1,50,1,351000,1,134,0,0,44,1 60,1,260,1,38,0,255000,2.2,132,0,1,45,1 95,1,371,0,30,0,461000,2,132,1,0,50,1 70,1,75,0,35,0,223000,2.7,138,1,1,54,0 60,1,607,0,40,0,216000,0.6,138,1,1,54,0 49,0,789,0,20,1,319000,1.1,136,1,1,55,1 72,0,364,1,20,1,254000,1.3,136,1,1,59,1 45,0,7702,1,25,1,390000,1,139,1,0,60,1 50,0,318,0,40,1,216000,2.3,131,0,0,60,1 55,0,109,0,35,0,254000,1.1,139,1,1,60,0 45,0,582,0,35,0,385000,1,145,1,0,61,1 45,0,582,0,80,0,263358.03,1.18,137,0,0,63,0 60,0,68,0,20,0,119000,2.9,127,1,1,64,1 42,1,250,1,15,0,213000,1.3,136,0,0,65,1 72,1,110,0,25,0,274000,1,140,1,1,65,1 70,0,161,0,25,0,244000,1.2,142,0,0,66,1 65,0,113,1,25,0,497000,1.83,135,1,0,67,1 41,0,148,0,40,0,374000,0.8,140,1,1,68,0 58,0,582,1,35,0,122000,0.9,139,1,1,71,0 85,0,5882,0,35,0,243000,1,132,1,1,72,1 65,0,224,1,50,0,149000,1.3,137,1,1,72,0 69,0,582,0,20,0,266000,1.2,134,1,1,73,1 60,1,47,0,20,0,204000,0.7,139,1,1,73,1 70,0,92,0,60,1,317000,0.8,140,0,1,74,0 42,0,102,1,40,0,237000,1.2,140,1,0,74,0 75,1,203,1,38,1,283000,0.6,131,1,1,74,0 55,0,336,0,45,1,324000,0.9,140,0,0,74,0 70,0,69,0,40,0,293000,1.7,136,0,0,75,0 67,0,582,0,50,0,263358.03,1.18,137,1,1,76,0 60,1,76,1,25,0,196000,2.5,132,0,0,77,1 79,1,55,0,50,1,172000,1.8,133,1,0,78,0 59,1,280,1,25,1,302000,1,141,0,0,78,1 51,0,78,0,50,0,406000,0.7,140,1,0,79,0 55,0,47,0,35,1,173000,1.1,137,1,0,79,0 65,1,68,1,60,1,304000,0.8,140,1,0,79,0 44,0,84,1,40,1,235000,0.7,139,1,0,79,0 57,1,115,0,25,1,181000,1.1,144,1,0,79,0 70,0,66,1,45,0,249000,0.8,136,1,1,80,0 60,0,897,1,45,0 1="" 0,="" ℎ??????="" find="" e(x)="" and="" var(x).="" q3="" (3pts).="" leptin="" (from="" the="" greek="" word="" leptos,="" meaning="" thin)="" is="" a="" 16-kda="" protein="" hormone="" that="" plays="" a="" crucial="" role="" in="" regulating="" energy="" intake="" and="" energy="" expenditure,="" including="" the="" regulation="" (decrease)="" of="" appetite="" and="" (increase)="" of="" metabolism.="" serum="" leptin="" concentrations="" can="" be="" measured="" in="" several="" ways.="" one="" approach="" uses="" a="" radioimmunoassay="" in="" venous="" blood="" samples="" (linco="" research="" inc.,="" st="" charles,="" mo).="" several="" studies="" have="" consistently="" found="" women="" to="" have="" higher="" serum="" leptin="" concentrations="" than="" do="" men.="" for="" example,="" among="" us="" adults="" across="" a="" broad="" age="" range,="" the="" mean="" serum="" leptin="" concentration="" in="" women="" is="" approximately="" normal="" (12.7??/?,="" (1.3??/?)2)="" and="" in="" men="" approximately="" normal="" (4.6??/?,="" (0.5??/?)2).="" 1)="" what="" is="" the="" probability="" that="" leptin="" concentration="" in="" a="" randomly="" selected="" us="" adult="" male="" exceeds="" 6="" 2)="" what="" proportion="" of="" us="" women="" have="" the="" concentration="" of="" leptin="" in="" the="" interval="" 12.7="" ±="" 2??/??="" 3)="" what="" interval,="" symmetric="" about="" the="" mean="" 12.7??/?,="" contains="" leptin="" concentrations="" of="" 95%="" of="" adult="" us="" women?="" q4="" (5pts).="" take="" a="" sample="" of="" size="" n="10" from="" the="" poisson="" distribution="" with="" mean="" =3="" and="" compute="" the="" sample="" mean.="" repeat="" this="" procedure="" 10,000="" times="" and="" plot="" the="" histogram="" of="" sample="" means.="" then="" repeat="" this="" procedure="" for="" n="100," 1000.="" is="" there="" any="" trend="" observed="" in="" the="" histogram="" of="" the="" sample="" mean="" as="" n="" increases?="" what="" is="" the="" property="" of="" the="" sampling="" distribution="" of="" the="" sample="" mean="" for="" n="1000?" what="" are="" the="" mean="" and="" variance="" of="" this="" distribution?="" q5="" (5pts).="" use="" the="" data="" in="" hw1="" to="" make="" the="" following="" statistical="" inferences="" about="" the="" following="" attributes="" in="" a="" general="" heart="" disease="" population="" where="" the="" data="" were="" generated.="" 1)="" construct="" the="" 95%="" confidence="" intervals="" for="" average="" creatinine="" phosphokinase="" (cpk)="" levels="" in="" deceased="" and="" alive="" patients="" in="" the="" population,="" respectively.="" 2)="" construct="" the="" 95%="" confidence="" intervals="" for="" variances="" of="" cpk="" levels="" in="" deceased="" and="" alive="" patients="" in="" the="" population,="" respectively.="" 3)="" construct="" the="" 95%="" confidence="" intervals="" for="" proportions="" of="" diabetes="" in="" deceased="" and="" alive="" patients="" in="" the="" population,="" respectively.="" justify="" the="" sampling="" distributions="" used="" for="" the="" calculation="" in="" (1)-(3).="" you="" may="" use="" the="" histograms="" of="" the="" cpk="" levels="" for="" 1)="" and="" 2).="" submission.="" submit="" a="" zip="" file="" (hw3_name.zip)="" on="" blackboard="" that="" includes:="" 1)="" both="" r="" notebook="" (.rmd)="" and="" rendered="" results="" (.html).="" your="" notebook="" should="" include="" some="" annotations="" for="" the="" scripts="" so="" that="" a="" grader="" knows="" what="" the="" script="" is="" for.="" 2)="" report="" document="" (.pdf)="" age,anaemia,creatinine_phosphokinase,diabetes,ejection_fraction,high_blood_pressure,platelets,serum_creatinine,serum_sodium,sex,smoking,time,death_event="" 75,0,582,0,20,1,265000,1.9,130,1,0,4,1="" 55,0,7861,0,38,0,263358.03,1.1,136,1,0,6,1="" 65,0,146,0,20,0,162000,1.3,129,1,1,7,1="" 50,1,111,0,20,0,210000,1.9,137,1,0,7,1="" 65,1,160,1,20,0,327000,2.7,116,0,0,8,1="" 90,1,47,0,40,1,204000,2.1,132,1,1,8,1="" 75,1,246,0,15,0,127000,1.2,137,1,0,10,1="" 60,1,315,1,60,0,454000,1.1,131,1,1,10,1="" 65,0,157,0,65,0,263358.03,1.5,138,0,0,10,1="" 80,1,123,0,35,1,388000,9.4,133,1,1,10,1="" 75,1,81,0,38,1,368000,4,131,1,1,10,1="" 62,0,231,0,25,1,253000,0.9,140,1,1,10,1="" 45,1,981,0,30,0,136000,1.1,137,1,0,11,1="" 50,1,168,0,38,1,276000,1.1,137,1,0,11,1="" 49,1,80,0,30,1,427000,1,138,0,0,12,0="" 82,1,379,0,50,0,47000,1.3,136,1,0,13,1="" 87,1,149,0,38,0,262000,0.9,140,1,0,14,1="" 45,0,582,0,14,0,166000,0.8,127,1,0,14,1="" 70,1,125,0,25,1,237000,1,140,0,0,15,1="" 48,1,582,1,55,0,87000,1.9,121,0,0,15,1="" 65,1,52,0,25,1,276000,1.3,137,0,0,16,0="" 65,1,128,1,30,1,297000,1.6,136,0,0,20,1="" 68,1,220,0,35,1,289000,0.9,140,1,1,20,1="" 53,0,63,1,60,0,368000,0.8,135,1,0,22,0="" 75,0,582,1,30,1,263358.03,1.83,134,0,0,23,1="" 80,0,148,1,38,0,149000,1.9,144,1,1,23,1="" 95,1,112,0,40,1,196000,1,138,0,0,24,1="" 70,0,122,1,45,1,284000,1.3,136,1,1,26,1="" 58,1,60,0,38,0,153000,5.8,134,1,0,26,1="" 82,0,70,1,30,0,200000,1.2,132,1,1,26,1="" 94,0,582,1,38,1,263358.03,1.83,134,1,0,27,1="" 85,0,23,0,45,0,360000,3,132,1,0,28,1="" 50,1,249,1,35,1,319000,1,128,0,0,28,1="" 50,1,159,1,30,0,302000,1.2,138,0,0,29,0="" 65,0,94,1,50,1,188000,1,140,1,0,29,1="" 69,0,582,1,35,0,228000,3.5,134,1,0,30,1="" 90,1,60,1,50,0,226000,1,134,1,0,30,1="" 82,1,855,1,50,1,321000,1,145,0,0,30,1="" 60,0,2656,1,30,0,305000,2.3,137,1,0,30,0="" 60,0,235,1,38,0,329000,3,142,0,0,30,1="" 70,0,582,0,20,1,263358.03,1.83,134,1,1,31,1="" 50,0,124,1,30,1,153000,1.2,136,0,1,32,1="" 70,0,571,1,45,1,185000,1.2,139,1,1,33,1="" 72,0,127,1,50,1,218000,1,134,1,0,33,0="" 60,1,588,1,60,0,194000,1.1,142,0,0,33,1="" 50,0,582,1,38,0,310000,1.9,135,1,1,35,1="" 51,0,1380,0,25,1,271000,0.9,130,1,0,38,1="" 60,0,582,1,38,1,451000,0.6,138,1,1,40,1="" 80,1,553,0,20,1,140000,4.4,133,1,0,41,1="" 57,1,129,0,30,0,395000,1,140,0,0,42,1="" 68,1,577,0,25,1,166000,1,138,1,0,43,1="" 53,1,91,0,20,1,418000,1.4,139,0,0,43,1="" 60,0,3964,1,62,0,263358.03,6.8,146,0,0,43,1="" 70,1,69,1,50,1,351000,1,134,0,0,44,1="" 60,1,260,1,38,0,255000,2.2,132,0,1,45,1="" 95,1,371,0,30,0,461000,2,132,1,0,50,1="" 70,1,75,0,35,0,223000,2.7,138,1,1,54,0="" 60,1,607,0,40,0,216000,0.6,138,1,1,54,0="" 49,0,789,0,20,1,319000,1.1,136,1,1,55,1="" 72,0,364,1,20,1,254000,1.3,136,1,1,59,1="" 45,0,7702,1,25,1,390000,1,139,1,0,60,1="" 50,0,318,0,40,1,216000,2.3,131,0,0,60,1="" 55,0,109,0,35,0,254000,1.1,139,1,1,60,0="" 45,0,582,0,35,0,385000,1,145,1,0,61,1="" 45,0,582,0,80,0,263358.03,1.18,137,0,0,63,0="" 60,0,68,0,20,0,119000,2.9,127,1,1,64,1="" 42,1,250,1,15,0,213000,1.3,136,0,0,65,1="" 72,1,110,0,25,0,274000,1,140,1,1,65,1="" 70,0,161,0,25,0,244000,1.2,142,0,0,66,1="" 65,0,113,1,25,0,497000,1.83,135,1,0,67,1="" 41,0,148,0,40,0,374000,0.8,140,1,1,68,0="" 58,0,582,1,35,0,122000,0.9,139,1,1,71,0="" 85,0,5882,0,35,0,243000,1,132,1,1,72,1="" 65,0,224,1,50,0,149000,1.3,137,1,1,72,0="" 69,0,582,0,20,0,266000,1.2,134,1,1,73,1="" 60,1,47,0,20,0,204000,0.7,139,1,1,73,1="" 70,0,92,0,60,1,317000,0.8,140,0,1,74,0="" 42,0,102,1,40,0,237000,1.2,140,1,0,74,0="" 75,1,203,1,38,1,283000,0.6,131,1,1,74,0="" 55,0,336,0,45,1,324000,0.9,140,0,0,74,0="" 70,0,69,0,40,0,293000,1.7,136,0,0,75,0="" 67,0,582,0,50,0,263358.03,1.18,137,1,1,76,0="" 60,1,76,1,25,0,196000,2.5,132,0,0,77,1="" 79,1,55,0,50,1,172000,1.8,133,1,0,78,0="" 59,1,280,1,25,1,302000,1,141,0,0,78,1="" 51,0,78,0,50,0,406000,0.7,140,1,0,79,0="" 55,0,47,0,35,1,173000,1.1,137,1,0,79,0="" 65,1,68,1,60,1,304000,0.8,140,1,0,79,0="" 44,0,84,1,40,1,235000,0.7,139,1,0,79,0="" 57,1,115,0,25,1,181000,1.1,144,1,0,79,0="" 70,0,66,1,45,0,249000,0.8,136,1,1,80,0=""> 1 0, ??ℎ?????? find e(x) and var(x). q3 (3pts). leptin (from the greek word leptos, meaning thin) is a 16-kda protein hormone that plays a crucial role in regulating energy intake and energy expenditure, including the regulation (decrease) of appetite and (increase) of metabolism. serum leptin concentrations can be measured in several ways. one approach uses a radioimmunoassay in venous blood samples (linco research inc., st charles, mo). several studies have consistently found women to have higher serum leptin concentrations than do men. for example, among us adults across a broad age range, the mean serum leptin concentration in women is approximately normal ?(12.7??/?, (1.3??/?)2) and in men approximately normal ?(4.6??/?, (0.5??/?)2). 1) what is the probability that leptin concentration in a randomly selected us adult male exceeds 6 ??/?? 2) what proportion of us women have the concentration of leptin in the interval 12.7 ± 2??/?? 3) what interval, symmetric about the mean 12.7??/?, contains leptin concentrations of 95% of adult us women? q4 (5pts). take a sample of size n=10 from the poisson distribution with mean ?=3 and compute the sample mean. repeat this procedure 10,000 times and plot the histogram of sample means. then repeat this procedure for n=100, 1000. is there any trend observed in the histogram of the sample mean as n increases? what is the property of the sampling distribution of the sample mean for n=1000? what are the mean and variance of this distribution? q5 (5pts). use the data in hw1 to make the following statistical inferences about the following attributes in a general heart disease population where the data were generated. 1) construct the 95% confidence intervals for average creatinine phosphokinase (cpk) levels in deceased and alive patients in the population, respectively. 2) construct the 95% confidence intervals for variances of cpk levels in deceased and alive patients in the population, respectively. 3) construct the 95% confidence intervals for proportions of diabetes in deceased and alive patients in the population, respectively. justify the sampling distributions used for the calculation in (1)-(3). you may use the histograms of the cpk levels for 1) and 2). submission. submit a zip file (hw3_name.zip) on blackboard that includes: 1) both r notebook (.rmd) and rendered results (.html). your notebook should include some annotations for the scripts so that a grader knows what the script is for. 2) report document (.pdf) age,anaemia,creatinine_phosphokinase,diabetes,ejection_fraction,high_blood_pressure,platelets,serum_creatinine,serum_sodium,sex,smoking,time,death_event 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