AST 231
All Lecture 231
lecture 02
set working directory
# "D:/2nd Year/AST 231" replace '/' by '\\'
setwd("D:\\2nd Year\\AST 231\\AST 231")
# or, Session >set working directory > to source file location
setwd("D:/2nd Year/AST 231/AST 231")read excel file
data<-read.csv("D:\\2nd Year\\AST 231\\AST 231\\EXCELL PRACTICE.csv")
View(data)income <- read.table("D:/2nd Year/AST 231/EXCELL PRACTICE.csv", header=T)
View(income)data2<-read.dta(““) data3<-read.spss(”“) data4<-read.table(”“,header = FALSE) data5<-read.csv(”“)
head(data1)
for build in R data
library("foreign")
data("mtcars")
mtcars## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1
## Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4
## Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2
## Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2
## Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4
## Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4
## Merc 450SE 16.4 8 275.8 180 3.07 4.070 17.40 0 0 3 3
## Merc 450SL 17.3 8 275.8 180 3.07 3.730 17.60 0 0 3 3
## Merc 450SLC 15.2 8 275.8 180 3.07 3.780 18.00 0 0 3 3
## Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4
## Lincoln Continental 10.4 8 460.0 215 3.00 5.424 17.82 0 0 3 4
## Chrysler Imperial 14.7 8 440.0 230 3.23 5.345 17.42 0 0 3 4
## Fiat 128 32.4 4 78.7 66 4.08 2.200 19.47 1 1 4 1
## Honda Civic 30.4 4 75.7 52 4.93 1.615 18.52 1 1 4 2
## Toyota Corolla 33.9 4 71.1 65 4.22 1.835 19.90 1 1 4 1
## Toyota Corona 21.5 4 120.1 97 3.70 2.465 20.01 1 0 3 1
## Dodge Challenger 15.5 8 318.0 150 2.76 3.520 16.87 0 0 3 2
## AMC Javelin 15.2 8 304.0 150 3.15 3.435 17.30 0 0 3 2
## Camaro Z28 13.3 8 350.0 245 3.73 3.840 15.41 0 0 3 4
## Pontiac Firebird 19.2 8 400.0 175 3.08 3.845 17.05 0 0 3 2
## Fiat X1-9 27.3 4 79.0 66 4.08 1.935 18.90 1 1 4 1
## Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.70 0 1 5 2
## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2
## Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.50 0 1 5 4
## Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.50 0 1 5 6
## Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.60 0 1 5 8
## Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.60 1 1 4 2data("iris")
iris## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 5.1 3.5 1.4 0.2 setosa
## 2 4.9 3.0 1.4 0.2 setosa
## 3 4.7 3.2 1.3 0.2 setosa
## 4 4.6 3.1 1.5 0.2 setosa
## 5 5.0 3.6 1.4 0.2 setosa
## 6 5.4 3.9 1.7 0.4 setosa
## 7 4.6 3.4 1.4 0.3 setosa
## 8 5.0 3.4 1.5 0.2 setosa
## 9 4.4 2.9 1.4 0.2 setosa
## 10 4.9 3.1 1.5 0.1 setosa
## 11 5.4 3.7 1.5 0.2 setosa
## 12 4.8 3.4 1.6 0.2 setosa
## 13 4.8 3.0 1.4 0.1 setosa
## 14 4.3 3.0 1.1 0.1 setosa
## 15 5.8 4.0 1.2 0.2 setosa
## 16 5.7 4.4 1.5 0.4 setosa
## 17 5.4 3.9 1.3 0.4 setosa
## 18 5.1 3.5 1.4 0.3 setosa
## 19 5.7 3.8 1.7 0.3 setosa
## 20 5.1 3.8 1.5 0.3 setosa
## 21 5.4 3.4 1.7 0.2 setosa
## 22 5.1 3.7 1.5 0.4 setosa
## 23 4.6 3.6 1.0 0.2 setosa
## 24 5.1 3.3 1.7 0.5 setosa
## 25 4.8 3.4 1.9 0.2 setosa
## 26 5.0 3.0 1.6 0.2 setosa
## 27 5.0 3.4 1.6 0.4 setosa
## 28 5.2 3.5 1.5 0.2 setosa
## 29 5.2 3.4 1.4 0.2 setosa
## 30 4.7 3.2 1.6 0.2 setosa
## 31 4.8 3.1 1.6 0.2 setosa
## 32 5.4 3.4 1.5 0.4 setosa
## 33 5.2 4.1 1.5 0.1 setosa
## 34 5.5 4.2 1.4 0.2 setosa
## 35 4.9 3.1 1.5 0.2 setosa
## 36 5.0 3.2 1.2 0.2 setosa
## 37 5.5 3.5 1.3 0.2 setosa
## 38 4.9 3.6 1.4 0.1 setosa
## 39 4.4 3.0 1.3 0.2 setosa
## 40 5.1 3.4 1.5 0.2 setosa
## 41 5.0 3.5 1.3 0.3 setosa
## 42 4.5 2.3 1.3 0.3 setosa
## 43 4.4 3.2 1.3 0.2 setosa
## 44 5.0 3.5 1.6 0.6 setosa
## 45 5.1 3.8 1.9 0.4 setosa
## 46 4.8 3.0 1.4 0.3 setosa
## 47 5.1 3.8 1.6 0.2 setosa
## 48 4.6 3.2 1.4 0.2 setosa
## 49 5.3 3.7 1.5 0.2 setosa
## 50 5.0 3.3 1.4 0.2 setosa
## 51 7.0 3.2 4.7 1.4 versicolor
## 52 6.4 3.2 4.5 1.5 versicolor
## 53 6.9 3.1 4.9 1.5 versicolor
## 54 5.5 2.3 4.0 1.3 versicolor
## 55 6.5 2.8 4.6 1.5 versicolor
## 56 5.7 2.8 4.5 1.3 versicolor
## 57 6.3 3.3 4.7 1.6 versicolor
## 58 4.9 2.4 3.3 1.0 versicolor
## 59 6.6 2.9 4.6 1.3 versicolor
## 60 5.2 2.7 3.9 1.4 versicolor
## 61 5.0 2.0 3.5 1.0 versicolor
## 62 5.9 3.0 4.2 1.5 versicolor
## 63 6.0 2.2 4.0 1.0 versicolor
## 64 6.1 2.9 4.7 1.4 versicolor
## 65 5.6 2.9 3.6 1.3 versicolor
## 66 6.7 3.1 4.4 1.4 versicolor
## 67 5.6 3.0 4.5 1.5 versicolor
## 68 5.8 2.7 4.1 1.0 versicolor
## 69 6.2 2.2 4.5 1.5 versicolor
## 70 5.6 2.5 3.9 1.1 versicolor
## 71 5.9 3.2 4.8 1.8 versicolor
## 72 6.1 2.8 4.0 1.3 versicolor
## 73 6.3 2.5 4.9 1.5 versicolor
## 74 6.1 2.8 4.7 1.2 versicolor
## 75 6.4 2.9 4.3 1.3 versicolor
## 76 6.6 3.0 4.4 1.4 versicolor
## 77 6.8 2.8 4.8 1.4 versicolor
## 78 6.7 3.0 5.0 1.7 versicolor
## 79 6.0 2.9 4.5 1.5 versicolor
## 80 5.7 2.6 3.5 1.0 versicolor
## 81 5.5 2.4 3.8 1.1 versicolor
## 82 5.5 2.4 3.7 1.0 versicolor
## 83 5.8 2.7 3.9 1.2 versicolor
## 84 6.0 2.7 5.1 1.6 versicolor
## 85 5.4 3.0 4.5 1.5 versicolor
## 86 6.0 3.4 4.5 1.6 versicolor
## 87 6.7 3.1 4.7 1.5 versicolor
## 88 6.3 2.3 4.4 1.3 versicolor
## 89 5.6 3.0 4.1 1.3 versicolor
## 90 5.5 2.5 4.0 1.3 versicolor
## 91 5.5 2.6 4.4 1.2 versicolor
## 92 6.1 3.0 4.6 1.4 versicolor
## 93 5.8 2.6 4.0 1.2 versicolor
## 94 5.0 2.3 3.3 1.0 versicolor
## 95 5.6 2.7 4.2 1.3 versicolor
## 96 5.7 3.0 4.2 1.2 versicolor
## 97 5.7 2.9 4.2 1.3 versicolor
## 98 6.2 2.9 4.3 1.3 versicolor
## 99 5.1 2.5 3.0 1.1 versicolor
## 100 5.7 2.8 4.1 1.3 versicolor
## 101 6.3 3.3 6.0 2.5 virginica
## 102 5.8 2.7 5.1 1.9 virginica
## 103 7.1 3.0 5.9 2.1 virginica
## 104 6.3 2.9 5.6 1.8 virginica
## 105 6.5 3.0 5.8 2.2 virginica
## 106 7.6 3.0 6.6 2.1 virginica
## 107 4.9 2.5 4.5 1.7 virginica
## 108 7.3 2.9 6.3 1.8 virginica
## 109 6.7 2.5 5.8 1.8 virginica
## 110 7.2 3.6 6.1 2.5 virginica
## 111 6.5 3.2 5.1 2.0 virginica
## 112 6.4 2.7 5.3 1.9 virginica
## 113 6.8 3.0 5.5 2.1 virginica
## 114 5.7 2.5 5.0 2.0 virginica
## 115 5.8 2.8 5.1 2.4 virginica
## 116 6.4 3.2 5.3 2.3 virginica
## 117 6.5 3.0 5.5 1.8 virginica
## 118 7.7 3.8 6.7 2.2 virginica
## 119 7.7 2.6 6.9 2.3 virginica
## 120 6.0 2.2 5.0 1.5 virginica
## 121 6.9 3.2 5.7 2.3 virginica
## 122 5.6 2.8 4.9 2.0 virginica
## 123 7.7 2.8 6.7 2.0 virginica
## 124 6.3 2.7 4.9 1.8 virginica
## 125 6.7 3.3 5.7 2.1 virginica
## 126 7.2 3.2 6.0 1.8 virginica
## 127 6.2 2.8 4.8 1.8 virginica
## 128 6.1 3.0 4.9 1.8 virginica
## 129 6.4 2.8 5.6 2.1 virginica
## 130 7.2 3.0 5.8 1.6 virginica
## 131 7.4 2.8 6.1 1.9 virginica
## 132 7.9 3.8 6.4 2.0 virginica
## 133 6.4 2.8 5.6 2.2 virginica
## 134 6.3 2.8 5.1 1.5 virginica
## 135 6.1 2.6 5.6 1.4 virginica
## 136 7.7 3.0 6.1 2.3 virginica
## 137 6.3 3.4 5.6 2.4 virginica
## 138 6.4 3.1 5.5 1.8 virginica
## 139 6.0 3.0 4.8 1.8 virginica
## 140 6.9 3.1 5.4 2.1 virginica
## 141 6.7 3.1 5.6 2.4 virginica
## 142 6.9 3.1 5.1 2.3 virginica
## 143 5.8 2.7 5.1 1.9 virginica
## 144 6.8 3.2 5.9 2.3 virginica
## 145 6.7 3.3 5.7 2.5 virginica
## 146 6.7 3.0 5.2 2.3 virginica
## 147 6.3 2.5 5.0 1.9 virginica
## 148 6.5 3.0 5.2 2.0 virginica
## 149 6.2 3.4 5.4 2.3 virginica
## 150 5.9 3.0 5.1 1.8 virginicarandom number genarate
set.seed(13)
x1<-rnorm(n=10,mean=10,sd =4)
x1## [1] 12.217308 8.878912 17.100653 10.749280 14.570105 11.662105 14.918026
## [8] 10.946719 8.538469 14.420577rbinom(n = 5,size = 1,prob = .5)## [1] 0 1 1 1 0rweibull(n = 6,shape = 5,scale = 6)## [1] 5.176618 7.686552 5.699531 6.125001 5.128995 5.821886lecture 03
calculate probability distribution (pdf)
dbinom(x=2,size=10,prob=.6)## [1] 0.01061683lecture 04
Example (cdf)
calculate p(1<X<=3)
#p(x<=3)
p3<-pbinom(q = 3,size = 10,prob = 0.6,lower.tail = T)
p3## [1] 0.05476188#p(x<=1)
p1<-pbinom(q = 1,size = 10,prob = 0.6,lower.tail = TRUE)
p1## [1] 0.001677722#p(1<X<=3)
p<-p3-p1
p## [1] 0.05308416with replacement sample
X=3,6,7,10,12 p=.1,.3,.2,.2,.2 draw 3 sample each time
a<-c(3,6,7,10,12)
p<-c(.1,.3,.2,.2,.2)
sample(x = a,size = 3,prob =p,replace = TRUE)## [1] 7 3 12without replacement sample
X=3,6,7,10,12 p=.1,.3,.2,.2,.2 draw 3 sample each time
a<-c(3,6,7,10,12)
p<-c(.1,.3,.2,.2,.2)
sample(x = a,size = 3,prob =p,replace = FALSE)## [1] 10 6 12set.seed(13)
s1<-sample(x = a,size = 500,prob =p,replace = TRUE)
mean(s1)## [1] 7.818var(s1)## [1] 7.720317# main for histogram title
hist(s1,main = "500 discrete draws")
lecture 05
Distribution of sample statistic
Find the distribution of mean & variance by drawing 100 sized sample from N(4,10) 1000 times
y<-rnorm(n = 100,mean = 4,sd = sqrt(10))
y## [1] 5.94095626 5.30480169 2.54885080 1.87122266 4.61049129 8.37320566
## [7] 4.20544476 11.28033382 -2.58916772 0.34077286 5.97264666 -0.81570326
## [13] -4.25688404 11.98972795 5.19499480 4.08995613 1.76734170 6.00741276
## [19] 7.69615887 3.02544085 1.77444953 -0.01607200 4.30224095 2.58782506
## [25] -1.40553936 -0.56071026 8.02611207 6.42184564 1.05783075 8.57910426
## [31] 2.37937416 11.38726222 6.07043166 3.55919294 2.40626779 4.18114778
## [37] -0.55329936 7.96817037 5.32811829 4.33508152 4.28438528 -3.52971177
## [43] 13.18859404 4.81202635 4.49419261 -0.06325470 2.58140991 -1.02677180
## [49] 0.02812929 3.66213085 -0.03137001 0.50945677 4.42764435 5.05412192
## [55] 0.71627485 4.69479016 5.25635014 8.61550432 0.31991258 -0.28487939
## [61] 7.39083076 7.38364497 3.47262499 7.52295955 -3.05692168 9.00954065
## [67] 0.54797634 5.50822404 5.05891439 4.52457307 5.07013784 6.06685399
## [73] 1.14483089 3.90957473 1.69709805 3.57744188 6.49235527 1.09596700
## [79] 8.27527031 5.77492594 8.18114451 7.01765112 6.77193608 6.15564925
## [85] 5.37052374 8.83560748 -2.53471118 6.77609417 8.53180195 4.30523765
## [91] -3.33215231 2.54383781 0.78827961 1.02200100 8.34790807 4.44940409
## [97] 5.50028130 -0.68467894 8.77848863 4.33078519m<-NULL
n<-NULL
for(i in 1:1000){
y<-rnorm(n = 100,mean = 4,sd = sqrt(10))
m[i]<-mean(y)
n[i]<-var(y)
}
hist(m)
hist(n)
N~(10,0.6) find the distribution of getting success,(p)
p<-NULL
for(i in 1:1000){
y<-rbinom(n = 100,size = 10,prob = 0.6)
m<-mean(y)
#note: mean=n*p or,p=mean/n
p[i]<-m/10
}
p## [1] 0.614 0.617 0.608 0.634 0.628 0.577 0.582 0.591 0.598 0.599 0.587 0.566
## [13] 0.594 0.603 0.613 0.616 0.597 0.577 0.595 0.615 0.596 0.594 0.604 0.580
## [25] 0.631 0.628 0.586 0.603 0.618 0.610 0.576 0.595 0.629 0.590 0.610 0.592
## [37] 0.615 0.581 0.582 0.590 0.583 0.633 0.599 0.622 0.587 0.616 0.595 0.621
## [49] 0.611 0.585 0.587 0.586 0.598 0.587 0.593 0.589 0.600 0.611 0.598 0.613
## [61] 0.594 0.590 0.599 0.595 0.581 0.595 0.583 0.599 0.628 0.613 0.613 0.602
## [73] 0.598 0.624 0.598 0.594 0.603 0.573 0.592 0.629 0.588 0.609 0.586 0.589
## [85] 0.582 0.566 0.595 0.594 0.576 0.598 0.614 0.598 0.601 0.585 0.605 0.616
## [97] 0.591 0.590 0.585 0.608 0.568 0.590 0.581 0.608 0.591 0.603 0.606 0.598
## [109] 0.600 0.604 0.591 0.596 0.615 0.580 0.586 0.594 0.616 0.627 0.606 0.578
## [121] 0.571 0.592 0.628 0.601 0.604 0.594 0.598 0.597 0.593 0.627 0.599 0.601
## [133] 0.605 0.602 0.584 0.622 0.606 0.627 0.593 0.591 0.601 0.591 0.595 0.611
## [145] 0.595 0.616 0.591 0.579 0.629 0.621 0.624 0.578 0.600 0.605 0.610 0.585
## [157] 0.591 0.593 0.606 0.584 0.574 0.572 0.615 0.597 0.619 0.578 0.582 0.597
## [169] 0.618 0.599 0.604 0.599 0.607 0.616 0.601 0.597 0.646 0.596 0.606 0.587
## [181] 0.592 0.608 0.628 0.601 0.598 0.584 0.598 0.613 0.589 0.584 0.568 0.592
## [193] 0.578 0.623 0.584 0.620 0.563 0.584 0.605 0.586 0.574 0.611 0.598 0.611
## [205] 0.633 0.602 0.583 0.595 0.633 0.607 0.598 0.600 0.614 0.601 0.590 0.571
## [217] 0.606 0.615 0.610 0.610 0.563 0.613 0.596 0.594 0.590 0.598 0.604 0.596
## [229] 0.606 0.583 0.586 0.598 0.585 0.590 0.604 0.632 0.604 0.601 0.565 0.608
## [241] 0.597 0.590 0.623 0.591 0.605 0.636 0.609 0.621 0.588 0.576 0.595 0.611
## [253] 0.582 0.578 0.636 0.590 0.612 0.580 0.610 0.600 0.596 0.605 0.598 0.578
## [265] 0.594 0.596 0.581 0.602 0.594 0.610 0.576 0.584 0.595 0.609 0.606 0.620
## [277] 0.606 0.584 0.564 0.598 0.566 0.584 0.606 0.605 0.590 0.607 0.601 0.599
## [289] 0.575 0.596 0.594 0.606 0.606 0.616 0.611 0.594 0.590 0.606 0.615 0.629
## [301] 0.581 0.567 0.589 0.616 0.578 0.565 0.584 0.578 0.584 0.608 0.579 0.589
## [313] 0.589 0.618 0.617 0.590 0.606 0.621 0.610 0.594 0.599 0.590 0.603 0.618
## [325] 0.598 0.614 0.605 0.581 0.591 0.590 0.567 0.604 0.606 0.603 0.575 0.589
## [337] 0.609 0.597 0.602 0.617 0.598 0.580 0.620 0.618 0.593 0.601 0.580 0.613
## [349] 0.615 0.616 0.614 0.636 0.639 0.582 0.603 0.616 0.594 0.597 0.561 0.578
## [361] 0.630 0.588 0.625 0.605 0.613 0.589 0.596 0.586 0.597 0.599 0.589 0.590
## [373] 0.610 0.568 0.616 0.633 0.615 0.578 0.614 0.603 0.596 0.587 0.592 0.611
## [385] 0.597 0.619 0.602 0.596 0.618 0.594 0.583 0.621 0.589 0.608 0.600 0.633
## [397] 0.597 0.616 0.624 0.584 0.616 0.625 0.595 0.570 0.592 0.605 0.578 0.633
## [409] 0.614 0.586 0.601 0.589 0.588 0.606 0.604 0.597 0.575 0.598 0.606 0.609
## [421] 0.584 0.630 0.596 0.586 0.576 0.609 0.627 0.626 0.577 0.610 0.601 0.640
## [433] 0.578 0.604 0.569 0.590 0.604 0.611 0.581 0.576 0.599 0.596 0.585 0.574
## [445] 0.586 0.606 0.597 0.601 0.618 0.603 0.590 0.600 0.606 0.598 0.596 0.622
## [457] 0.582 0.599 0.617 0.590 0.598 0.607 0.609 0.595 0.596 0.592 0.603 0.601
## [469] 0.608 0.595 0.609 0.590 0.582 0.585 0.602 0.567 0.593 0.628 0.602 0.620
## [481] 0.622 0.596 0.597 0.593 0.611 0.594 0.595 0.609 0.612 0.578 0.597 0.602
## [493] 0.622 0.601 0.641 0.599 0.575 0.592 0.581 0.606 0.568 0.622 0.598 0.613
## [505] 0.614 0.577 0.581 0.581 0.596 0.590 0.596 0.614 0.610 0.616 0.614 0.594
## [517] 0.600 0.594 0.603 0.588 0.613 0.613 0.581 0.624 0.618 0.595 0.586 0.619
## [529] 0.647 0.591 0.606 0.595 0.594 0.580 0.591 0.581 0.622 0.590 0.599 0.574
## [541] 0.609 0.605 0.619 0.598 0.602 0.584 0.599 0.603 0.573 0.598 0.611 0.589
## [553] 0.565 0.616 0.642 0.613 0.644 0.606 0.602 0.613 0.606 0.626 0.604 0.585
## [565] 0.590 0.580 0.581 0.588 0.603 0.600 0.587 0.594 0.617 0.595 0.596 0.619
## [577] 0.619 0.609 0.599 0.607 0.586 0.610 0.599 0.592 0.632 0.584 0.580 0.613
## [589] 0.603 0.588 0.579 0.599 0.602 0.597 0.582 0.615 0.584 0.617 0.610 0.610
## [601] 0.588 0.573 0.580 0.589 0.589 0.580 0.595 0.606 0.584 0.593 0.591 0.592
## [613] 0.615 0.613 0.609 0.589 0.590 0.594 0.583 0.597 0.569 0.588 0.580 0.599
## [625] 0.595 0.618 0.593 0.613 0.630 0.593 0.589 0.626 0.602 0.616 0.582 0.584
## [637] 0.582 0.597 0.595 0.606 0.628 0.634 0.589 0.586 0.589 0.583 0.597 0.601
## [649] 0.610 0.606 0.608 0.635 0.622 0.593 0.606 0.593 0.601 0.600 0.598 0.626
## [661] 0.578 0.596 0.581 0.589 0.599 0.593 0.605 0.588 0.600 0.606 0.591 0.592
## [673] 0.625 0.617 0.619 0.609 0.611 0.591 0.592 0.564 0.601 0.599 0.611 0.606
## [685] 0.611 0.571 0.600 0.608 0.594 0.597 0.589 0.589 0.611 0.622 0.583 0.595
## [697] 0.639 0.590 0.589 0.601 0.603 0.609 0.581 0.587 0.602 0.607 0.625 0.600
## [709] 0.607 0.600 0.623 0.591 0.580 0.618 0.614 0.579 0.592 0.578 0.583 0.604
## [721] 0.591 0.596 0.598 0.595 0.620 0.604 0.607 0.605 0.582 0.599 0.600 0.616
## [733] 0.593 0.619 0.561 0.591 0.615 0.599 0.618 0.592 0.570 0.579 0.600 0.616
## [745] 0.558 0.586 0.587 0.602 0.578 0.609 0.592 0.602 0.602 0.603 0.617 0.567
## [757] 0.605 0.617 0.597 0.608 0.620 0.603 0.617 0.600 0.584 0.577 0.608 0.593
## [769] 0.617 0.615 0.612 0.572 0.600 0.588 0.607 0.605 0.574 0.583 0.597 0.602
## [781] 0.610 0.575 0.618 0.605 0.619 0.579 0.608 0.597 0.587 0.613 0.612 0.578
## [793] 0.585 0.592 0.591 0.635 0.628 0.604 0.592 0.586 0.598 0.585 0.596 0.598
## [805] 0.613 0.599 0.596 0.598 0.624 0.612 0.621 0.615 0.574 0.598 0.603 0.589
## [817] 0.605 0.586 0.611 0.578 0.561 0.558 0.613 0.599 0.598 0.588 0.609 0.587
## [829] 0.588 0.595 0.606 0.582 0.632 0.582 0.583 0.584 0.598 0.602 0.611 0.569
## [841] 0.631 0.613 0.581 0.596 0.568 0.586 0.596 0.614 0.615 0.605 0.601 0.582
## [853] 0.594 0.571 0.604 0.629 0.599 0.569 0.600 0.610 0.603 0.602 0.589 0.594
## [865] 0.614 0.597 0.588 0.624 0.602 0.589 0.564 0.607 0.575 0.609 0.589 0.580
## [877] 0.602 0.610 0.600 0.591 0.606 0.607 0.567 0.592 0.603 0.606 0.599 0.590
## [889] 0.553 0.609 0.617 0.591 0.593 0.610 0.579 0.593 0.602 0.566 0.599 0.592
## [901] 0.599 0.625 0.600 0.600 0.596 0.611 0.602 0.596 0.625 0.610 0.588 0.611
## [913] 0.624 0.592 0.595 0.594 0.601 0.574 0.606 0.601 0.627 0.608 0.612 0.607
## [925] 0.602 0.577 0.616 0.583 0.608 0.598 0.587 0.575 0.595 0.598 0.592 0.591
## [937] 0.618 0.595 0.588 0.589 0.598 0.609 0.613 0.595 0.619 0.585 0.575 0.604
## [949] 0.585 0.624 0.608 0.595 0.618 0.587 0.601 0.592 0.632 0.567 0.591 0.603
## [961] 0.582 0.588 0.594 0.596 0.574 0.604 0.589 0.604 0.595 0.583 0.608 0.579
## [973] 0.591 0.602 0.592 0.597 0.589 0.613 0.609 0.622 0.589 0.623 0.605 0.601
## [985] 0.614 0.611 0.597 0.597 0.590 0.593 0.591 0.602 0.590 0.616 0.618 0.574
## [997] 0.627 0.603 0.590 0.596hist(p)
### fairness of coin fairness of a coin (H0: coin not fair; H1: coin
fair)
y<-NULL
for (i in 1:1000){
s<-rbinom(1000,1,0.6)
y[i]<-sum(s)
}
lw<-quantile(y,0.025)
lw## 2.5%
## 570up<-quantile(y,0.975)
up## 97.5%
## 630.025#P(H1)
p_H1<-sum (570<y & y<629)/length(y)
p_H1## [1] 0.938#P(H0)
p_value<-1-p_H1
p_value## [1] 0.062lecture 06
Generate random number
1.Draw a sample of size 200 from N (5,8) distribution.Find the percentage of sample above 6.
set.seed(13)
y<-rnorm(200,5,sqrt(8))
sum(y>6)/200*100## [1] 35.5simulate
2.A fair coin flipped 10 times. Simulate thisprocess 1000 times and find how many times you get an unequal number of headsand tails.
set.seed(13)
y<-NULL
for(i in 1:1000){
x<-rbinom(10,1,.5)
y[i]<-sum(x)
}
sum(y!=5)## [1] 7513.Generate a sample of size 50 from poisson (10). Simulate this process 5000 times and find the probability of estimated lambda greater than 10.
set.seed(13)
y<-NULL
for(i in 1:5000){
x<-rpois(n = 50,lambda = 10)
y[i]<-mean(x)
}
sum(y>10)/5000## [1] 0.4846lecture 07 (same 6)
1.Draw a sample of size 200 from N (5,8) distribution.Find the percentage of sample above 6.
set.seed(123)
y<-rnorm(200,5,sqrt(8))
sum(y>6)/200*100## [1] 33.52.A fair coin flipped 10 times. Simulate this process 1000 times and find how many times you get an unequal number of heads and tails.
set.seed(12)
rbinom(n = 10,size = 1,.5)## [1] 0 1 1 0 0 0 0 1 0 0y<-NULL
for(i in 1:1000){
y[i]<-sum(rbinom(n = 10,size = 1,.5))
}
sum(y!=5)## [1] 764Generate a sample of size 50 from poisson (10).Simulate this process 5000 times and find the probability of estimated lambda greater than 10.
set.seed(123)
x<-NULL
for(i in 1:5000){
x[i]<-mean(rpois(50,10))
}
sum(x>10)/5000## [1] 0.493lecture 8
UNBIASEDNESS ex-1
Unbiasedness of sample mean
# Expected Value of sample equals to parameter value
population<-c(14,12,14,17,18,19,20,21,23,24)
length(population)## [1] 10mean(population)## [1] 18.2all_sample<-combn(x = population,m = 3)
all_sample## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,] 14 14 14 14 14 14 14 14 14 14 14 14 14 14
## [2,] 12 12 12 12 12 12 12 12 14 14 14 14 14 14
## [3,] 14 17 18 19 20 21 23 24 17 18 19 20 21 23
## [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
## [1,] 14 14 14 14 14 14 14 14 14 14 14 14
## [2,] 14 17 17 17 17 17 17 18 18 18 18 18
## [3,] 24 18 19 20 21 23 24 19 20 21 23 24
## [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
## [1,] 14 14 14 14 14 14 14 14 14 14 12 12
## [2,] 19 19 19 19 20 20 20 21 21 23 14 14
## [3,] 20 21 23 24 21 23 24 23 24 24 17 18
## [,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
## [1,] 12 12 12 12 12 12 12 12 12 12 12 12
## [2,] 14 14 14 14 14 17 17 17 17 17 17 18
## [3,] 19 20 21 23 24 18 19 20 21 23 24 19
## [,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61] [,62]
## [1,] 12 12 12 12 12 12 12 12 12 12 12 12
## [2,] 18 18 18 18 19 19 19 19 20 20 20 21
## [3,] 20 21 23 24 20 21 23 24 21 23 24 23
## [,63] [,64] [,65] [,66] [,67] [,68] [,69] [,70] [,71] [,72] [,73] [,74]
## [1,] 12 12 14 14 14 14 14 14 14 14 14 14
## [2,] 21 23 17 17 17 17 17 17 18 18 18 18
## [3,] 24 24 18 19 20 21 23 24 19 20 21 23
## [,75] [,76] [,77] [,78] [,79] [,80] [,81] [,82] [,83] [,84] [,85] [,86]
## [1,] 14 14 14 14 14 14 14 14 14 14 14 17
## [2,] 18 19 19 19 19 20 20 20 21 21 23 18
## [3,] 24 20 21 23 24 21 23 24 23 24 24 19
## [,87] [,88] [,89] [,90] [,91] [,92] [,93] [,94] [,95] [,96] [,97] [,98]
## [1,] 17 17 17 17 17 17 17 17 17 17 17 17
## [2,] 18 18 18 18 19 19 19 19 20 20 20 21
## [3,] 20 21 23 24 20 21 23 24 21 23 24 23
## [,99] [,100] [,101] [,102] [,103] [,104] [,105] [,106] [,107] [,108]
## [1,] 17 17 18 18 18 18 18 18 18 18
## [2,] 21 23 19 19 19 19 20 20 20 21
## [3,] 24 24 20 21 23 24 21 23 24 23
## [,109] [,110] [,111] [,112] [,113] [,114] [,115] [,116] [,117] [,118]
## [1,] 18 18 19 19 19 19 19 19 20 20
## [2,] 21 23 20 20 20 21 21 23 21 21
## [3,] 24 24 21 23 24 23 24 24 23 24
## [,119] [,120]
## [1,] 20 21
## [2,] 23 23
## [3,] 24 24dim(all_sample)## [1] 3 120all_mean<-apply(all_sample, 2, mean)
all_mean## [1] 13.33333 14.33333 14.66667 15.00000 15.33333 15.66667 16.33333 16.66667
## [9] 15.00000 15.33333 15.66667 16.00000 16.33333 17.00000 17.33333 16.33333
## [17] 16.66667 17.00000 17.33333 18.00000 18.33333 17.00000 17.33333 17.66667
## [25] 18.33333 18.66667 17.66667 18.00000 18.66667 19.00000 18.33333 19.00000
## [33] 19.33333 19.33333 19.66667 20.33333 14.33333 14.66667 15.00000 15.33333
## [41] 15.66667 16.33333 16.66667 15.66667 16.00000 16.33333 16.66667 17.33333
## [49] 17.66667 16.33333 16.66667 17.00000 17.66667 18.00000 17.00000 17.33333
## [57] 18.00000 18.33333 17.66667 18.33333 18.66667 18.66667 19.00000 19.66667
## [65] 16.33333 16.66667 17.00000 17.33333 18.00000 18.33333 17.00000 17.33333
## [73] 17.66667 18.33333 18.66667 17.66667 18.00000 18.66667 19.00000 18.33333
## [81] 19.00000 19.33333 19.33333 19.66667 20.33333 18.00000 18.33333 18.66667
## [89] 19.33333 19.66667 18.66667 19.00000 19.66667 20.00000 19.33333 20.00000
## [97] 20.33333 20.33333 20.66667 21.33333 19.00000 19.33333 20.00000 20.33333
## [105] 19.66667 20.33333 20.66667 20.66667 21.00000 21.66667 20.00000 20.66667
## [113] 21.00000 21.00000 21.33333 22.00000 21.33333 21.66667 22.33333 22.66667# mean of all sample mean
mean(all_mean)## [1] 18.2Unbiasedness of sample variance
var(population)## [1] 15.95556(1/10)*sum((population-18.2)^2)## [1] 14.36(1/9)*sum((population-18.2)^2)## [1] 15.95556all_var<-apply(all_sample, 2, var)
all_var## [1] 1.333333 6.333333 9.333333 13.000000 17.333333 22.333333 34.333333
## [8] 41.333333 3.000000 5.333333 8.333333 12.000000 16.333333 27.000000
## [15] 33.333333 4.333333 6.333333 9.000000 12.333333 21.000000 26.333333
## [22] 7.000000 9.333333 12.333333 20.333333 25.333333 10.333333 13.000000
## [29] 20.333333 25.000000 14.333333 21.000000 25.333333 22.333333 26.333333
## [36] 30.333333 6.333333 9.333333 13.000000 17.333333 22.333333 34.333333
## [43] 41.333333 10.333333 13.000000 16.333333 20.333333 30.333333 36.333333
## [50] 14.333333 17.333333 21.000000 30.333333 36.000000 19.000000 22.333333
## [57] 31.000000 36.333333 24.333333 32.333333 37.333333 34.333333 39.000000
## [64] 44.333333 4.333333 6.333333 9.000000 12.333333 21.000000 26.333333
## [71] 7.000000 9.333333 12.333333 20.333333 25.333333 10.333333 13.000000
## [78] 20.333333 25.000000 14.333333 21.000000 25.333333 22.333333 26.333333
## [85] 30.333333 1.000000 2.333333 4.333333 10.333333 14.333333 2.333333
## [92] 4.000000 9.333333 13.000000 4.333333 9.000000 12.333333 9.333333
## [99] 12.333333 14.333333 1.000000 2.333333 7.000000 10.333333 2.333333
## [106] 6.333333 9.333333 6.333333 9.000000 10.333333 1.000000 4.333333
## [113] 7.000000 4.000000 6.333333 7.000000 2.333333 4.333333 4.333333
## [120] 2.333333# mean of all sample variance
mean(all_var)## [1] 15.95556lecture 9
UNBIASEDNESS ex-2
fil<-read.csv(file = "pop.csv",header = T)
fil## x y z
## 1 0 -40.9567603 0
## 2 0 -31.8173316 1
## 3 0 107.1450810 0
## 4 1 -97.7407109 0
## 5 1 -117.0041975 0
## 6 1 179.2802790 0
## 7 1 -24.1607227 0
## 8 0 40.5572968 1
## 9 1 151.2887414 0
## 10 0 124.8838923 0
## 11 0 62.2545652 0
## 12 1 -63.9260760 0
## 13 0 -145.8464928 0
## 14 1 -2.7477640 0
## 15 0 -44.0767845 0
## 16 0 -13.0245172 2
## 17 1 38.2127378 0
## 18 0 167.9366431 1
## 19 1 93.6051624 1
## 20 0 147.4111576 0
## 21 1 -20.6316656 0
## 22 1 -56.5421425 0
## 23 1 209.4553604 0
## 24 0 232.3950901 1
## 25 1 83.8486289 0
## 26 1 -73.4998126 0
## 27 1 -50.1512328 0
## 28 1 -42.7356755 1
## 29 1 -49.3725404 0
## 30 1 -76.7139685 0
## 31 0 69.3881027 0
## 32 0 53.8185649 0
## 33 0 -21.4893204 0
## 34 1 -19.8374681 0
## 35 0 -49.2559555 0
## 36 0 20.8802796 0
## 37 0 54.6910402 0
## 38 0 -20.8925015 0
## 39 0 -20.2874362 0
## 40 1 -37.1932507 0
## 41 1 23.9101224 1
## 42 0 211.2839941 0
## 43 0 34.8629143 0
## 44 1 -76.6871949 3
## 45 0 112.9149960 0
## 46 1 -1.2402290 0
## 47 1 33.4339419 0
## 48 1 94.7919221 1
## 49 1 -21.9887931 1
## 50 1 -59.1032553 4
## 51 1 -219.2468873 0
## 52 0 83.1585027 0
## 53 1 76.1579654 0
## 54 0 -78.5099351 2
## 55 0 -7.0310764 0
## 56 0 87.7558202 0
## 57 1 -28.4126644 1
## 58 1 -148.9555310 2
## 59 0 107.1612030 0
## 60 0 -38.5885637 3
## 61 1 -56.3538732 0
## 62 1 158.2385502 0
## 63 1 -107.1282298 0
## 64 0 120.3250546 0
## 65 0 137.2432874 0
## 66 0 0.1859710 0
## 67 1 -10.6279767 0
## 68 0 -33.4176216 0
## 69 0 194.9965949 0
## 70 0 -56.2739201 0
## 71 1 23.2381664 0
## 72 1 144.7663063 0
## 73 1 -3.9280043 0
## 74 0 -6.9589154 0
## 75 1 62.7957178 0
## 76 0 108.1653836 0
## 77 0 87.5594111 0
## 78 0 -112.8465649 0
## 79 1 24.1107186 0
## 80 0 -22.3333949 0
## 81 0 84.7291927 0
## 82 0 43.1858067 0
## 83 1 213.4802674 0
## 84 1 36.8102713 0
## 85 0 -59.5843004 0
## 86 0 83.8387836 0
## 87 1 -14.2493291 1
## 88 0 -40.4721434 0
## 89 1 132.4941228 0
## 90 0 37.3857462 2
## 91 1 -36.3075215 0
## 92 1 -69.5186777 0
## 93 1 0.3291253 1
## 94 1 128.2298958 0
## 95 1 178.3789648 1
## 96 1 -50.9804133 0
## 97 1 -54.0817081 0
## 98 0 165.9711847 1
## 99 1 5.8837937 0
## 100 1 63.8328282 1View(fil)
population<-fil$x+fil$y+fil$z
population## [1] -40.956760 -30.817332 107.145081 -96.740711 -116.004198 180.280279
## [7] -23.160723 41.557297 152.288741 124.883892 62.254565 -62.926076
## [13] -145.846493 -1.747764 -44.076785 -11.024517 39.212738 168.936643
## [19] 95.605162 147.411158 -19.631666 -55.542142 210.455360 233.395090
## [25] 84.848629 -72.499813 -49.151233 -40.735676 -48.372540 -75.713968
## [31] 69.388103 53.818565 -21.489320 -18.837468 -49.255955 20.880280
## [37] 54.691040 -20.892501 -20.287436 -36.193251 25.910122 211.283994
## [43] 34.862914 -72.687195 112.914996 -0.240229 34.433942 96.791922
## [49] -19.988793 -54.103255 -218.246887 83.158503 77.157965 -76.509935
## [55] -7.031076 87.755820 -26.412664 -145.955531 107.161203 -35.588564
## [61] -55.353873 159.238550 -106.128230 120.325055 137.243287 0.185971
## [67] -9.627977 -33.417622 194.996595 -56.273920 24.238166 145.766306
## [73] -2.928004 -6.958915 63.795718 108.165384 87.559411 -112.846565
## [79] 25.110719 -22.333395 84.729193 43.185807 214.480267 37.810271
## [85] -59.584300 83.838784 -12.249329 -40.472143 133.494123 39.385746
## [91] -35.307521 -68.518678 2.329125 129.229896 180.378965 -49.980413
## [97] -53.081708 166.971185 6.883794 65.832828mean(population)## [1] 24.89928var(population)## [1] 8351.911all_sample<-combn(population,3)
all_mean<-apply(all_sample, 2, mean)
mean(all_mean)## [1] 24.89928all_var<-apply(all_sample, 2, var)
mean(all_var)## [1] 8351.911#for pc
#setwd("D:/")
#data_set<-read.csv("pop.csv")
#View(data_set)UNBIASEDNESS ex-3
when number of sample too large for calculation .we take some number of sample from whole sample set
fil<-read.csv(file = "pop.csv",header = T)
#population<-fil$x+fil$y+fil$z
population<-exp(fil$y)-2*fil$x+3*fil$z
population## [1] 1.631943e-18 3.000000e+00 3.408140e+46 -2.000000e+00 -2.000000e+00
## [6] 7.251634e+77 -2.000000e+00 4.109701e+17 5.056681e+65 1.723397e+54
## [11] 1.088464e+27 -2.000000e+00 4.567441e-64 -1.935929e+00 7.206024e-20
## [16] 6.000002e+00 3.940771e+16 8.589293e+72 4.489578e+40 1.046773e+64
## [21] -2.000000e+00 -2.000000e+00 9.232243e+90 8.470426e+100 2.600141e+36
## [26] -2.000000e+00 -2.000000e+00 1.000000e+00 -2.000000e+00 -2.000000e+00
## [31] 1.364175e+30 2.361053e+23 4.648435e-10 -2.000000e+00 4.058919e-22
## [36] 1.170012e+09 5.649591e+23 8.443099e-10 1.546245e-09 -2.000000e+00
## [41] 2.421220e+10 5.747418e+91 1.382838e+15 7.000000e+00 1.092345e+49
## [46] -1.710682e+00 3.312657e+14 1.470987e+41 1.000000e+00 1.000000e+01
## [51] -2.000000e+00 1.304004e+36 1.188459e+33 6.000000e+00 8.839797e-04
## [56] 1.293804e+38 1.000000e+00 4.000000e+00 3.463531e+46 9.000000e+00
## [61] -2.000000e+00 5.273867e+68 -2.000000e+00 1.805125e+52 4.017930e+59
## [66] 1.204387e+00 -1.999976e+00 3.068395e-15 4.852272e+84 3.635357e-25
## [71] 1.236538e+10 7.433751e+62 -1.980317e+00 9.501266e-04 1.869966e+27
## [76] 9.454297e+46 1.063088e+38 9.803013e-50 2.959048e+10 1.998618e-10
## [81] 6.272213e+36 5.693214e+18 5.167758e+92 9.693882e+15 1.326988e-26
## [86] 2.574667e+36 1.000001e+00 2.649546e-18 3.479096e+57 1.723548e+16
## [91] -2.000000e+00 -2.000000e+00 2.389752e+00 4.892560e+55 2.944422e+77
## [96] -2.000000e+00 -2.000000e+00 1.203288e+72 3.571692e+02 5.275275e+27mean(population)## [1] 8.470426e+98var(population)## [1] 7.174812e+199# 1st way
some_sample<-replicate(n = 10000,expr = sample(population,size = 5,replace = F))
some_mean<-apply(some_sample, 2, mean)
mean(some_mean)## [1] 8.978652e+98# 2nd way
x<-replicate(10000,mean(sample(population,size = 5,replace = F)))
mean(x)## [1] 8.605953e+98unbiasness of mean
set.seed(13)
population<-rexp(n = 30,rate = 0.9)
# mean in unbiased estimator
mean(population)## [1] 1.043536all_sample<-combn(population,4)
all_mean<-apply(all_sample, 2, mean)
mean(all_mean)## [1] 1.043536MLE
library(MASS)
#View(birthwt)
x<-birthwt$bwt
mean(x)## [1] 2944.587var(x)## [1] 531753.5fun<-function(par){
-sum (dnorm(x,par[1],sqrt(par[2]),log=T))
}
optim(par=c(2944.587,531753.5),fun)## $par
## [1] 2944.646 529134.694
##
## $value
## [1] 1513.56
##
## $counts
## function gradient
## 53 NA
##
## $convergence
## [1] 0
##
## $message
## NULLlecture 10
x is a sample from normal distribution .Find MLE for mean & variance.
MLE Of exponaltial distribution
## 1st Way
set.seed(13)
x<- rexp(100,4)
fun<-function(t){
sum(dexp(x = x,rate = t,log = T))
}
optimize(fun,interval = c(0,100),maximum = T)## $maximum
## [1] 4.471106
##
## $objective
## [1] 49.76359## 2nd way
set.seed(13)
x<- rexp(10000,4)
fun<-function(par){
-sum(dexp(x = x,rate = par[1],log = T))
}
optim(par = c(4),fn = fun)## Warning in optim(par = c(4), fn = fun): one-dimensional optimization by Nelder-Mead is unreliable:
## use "Brent" or optimize() directly## $par
## [1] 4.025781
##
## $value
## [1] -3927.997
##
## $counts
## function gradient
## 20 NA
##
## $convergence
## [1] 0
##
## $message
## NULLMLE of binomial dist.(coin toss)
suppose 100 toss
## 1st way
x<-rbinom(n = 10000,size = 1,prob = 0.5)
fun<-function(t){
sum(dbinom(x = x,size = 1,prob = t,log = T))
}
mle<-optimize(fun,interval = c(0,1),maximum = T)
mle## $maximum
## [1] 0.4966905
##
## $objective
## [1] -6931.254## 2nd way
x<-rbinom(n = 10000,size = 1,prob = 0.5)
fun<-function(par){
-sum(dbinom(x = x,size = 1,prob = par[1],log = T))
}
optim(par = c(.5),fn = fun)## Warning in optim(par = c(0.5), fn = fun): one-dimensional optimization by Nelder-Mead is unreliable:
## use "Brent" or optimize() directly## $par
## [1] 0.5003906
##
## $value
## [1] 6931.472
##
## $counts
## function gradient
## 16 NA
##
## $convergence
## [1] 0
##
## $message
## NULLMLE of Normal distribution
x<-rnorm(10000,mean=0,sd=1)
fun<-function(par){
-sum(dnorm(x = x,mean = par[1],sd = par[2],log = T))
}
mle<-optim(par = c(0,1),fun)
mle## $par
## [1] 0.02293321 0.99865517
##
## $value
## [1] 14176.13
##
## $counts
## function gradient
## 43 NA
##
## $convergence
## [1] 0
##
## $message
## NULLlecture 11
fairness of coin
A tosses a coin 100 times and gets 70 heads. Check whether the coin is fair or not. H0: P(H)=0.5 H1: P(H)=!0.5
y<-NULL
for (i in 1:1000){
x<-rbinom(100,1,0.5)
y[i]<-sum(x)/100
}
y## [1] 0.46 0.60 0.50 0.58 0.55 0.39 0.40 0.43 0.53 0.54 0.53 0.51 0.53 0.42
## [15] 0.50 0.47 0.52 0.53 0.44 0.51 0.52 0.49 0.46 0.56 0.46 0.45 0.51 0.56
## [29] 0.43 0.45 0.52 0.53 0.40 0.61 0.56 0.50 0.60 0.54 0.52 0.56 0.45 0.50
## [43] 0.55 0.46 0.50 0.49 0.47 0.52 0.60 0.54 0.50 0.56 0.50 0.43 0.55 0.45
## [57] 0.53 0.57 0.49 0.43 0.51 0.50 0.47 0.54 0.46 0.47 0.52 0.46 0.44 0.52
## [71] 0.50 0.55 0.42 0.45 0.40 0.47 0.48 0.49 0.46 0.43 0.47 0.47 0.53 0.51
## [85] 0.49 0.50 0.56 0.47 0.49 0.47 0.48 0.53 0.40 0.50 0.50 0.48 0.40 0.48
## [99] 0.53 0.47 0.57 0.49 0.53 0.50 0.54 0.56 0.47 0.54 0.45 0.55 0.54 0.37
## [113] 0.50 0.58 0.44 0.50 0.47 0.50 0.53 0.49 0.48 0.50 0.30 0.58 0.52 0.40
## [127] 0.52 0.40 0.51 0.52 0.43 0.57 0.40 0.50 0.54 0.57 0.49 0.48 0.48 0.46
## [141] 0.53 0.47 0.55 0.49 0.47 0.52 0.56 0.53 0.51 0.54 0.48 0.49 0.48 0.47
## [155] 0.52 0.50 0.49 0.49 0.58 0.62 0.46 0.46 0.56 0.46 0.59 0.54 0.54 0.49
## [169] 0.52 0.54 0.48 0.44 0.51 0.49 0.54 0.49 0.53 0.49 0.43 0.53 0.57 0.46
## [183] 0.53 0.42 0.49 0.45 0.60 0.46 0.60 0.47 0.54 0.49 0.51 0.49 0.39 0.54
## [197] 0.54 0.43 0.58 0.47 0.42 0.60 0.50 0.58 0.59 0.55 0.45 0.47 0.46 0.52
## [211] 0.57 0.48 0.52 0.50 0.60 0.52 0.53 0.50 0.50 0.48 0.47 0.53 0.50 0.48
## [225] 0.48 0.58 0.46 0.57 0.51 0.52 0.58 0.53 0.51 0.36 0.51 0.55 0.48 0.45
## [239] 0.49 0.44 0.48 0.42 0.61 0.45 0.49 0.53 0.46 0.52 0.49 0.52 0.52 0.57
## [253] 0.51 0.55 0.47 0.47 0.51 0.48 0.58 0.52 0.51 0.46 0.58 0.59 0.48 0.49
## [267] 0.45 0.55 0.45 0.49 0.34 0.40 0.58 0.48 0.40 0.45 0.39 0.56 0.49 0.51
## [281] 0.42 0.46 0.49 0.47 0.45 0.41 0.51 0.51 0.54 0.53 0.55 0.55 0.56 0.54
## [295] 0.42 0.48 0.56 0.55 0.50 0.52 0.57 0.43 0.54 0.47 0.46 0.48 0.44 0.50
## [309] 0.57 0.55 0.50 0.48 0.49 0.48 0.42 0.55 0.45 0.55 0.52 0.46 0.43 0.49
## [323] 0.49 0.39 0.49 0.47 0.48 0.56 0.51 0.56 0.48 0.47 0.55 0.43 0.59 0.47
## [337] 0.55 0.52 0.46 0.55 0.47 0.50 0.54 0.54 0.46 0.62 0.57 0.49 0.43 0.53
## [351] 0.52 0.41 0.46 0.45 0.53 0.54 0.61 0.54 0.55 0.49 0.53 0.49 0.44 0.46
## [365] 0.56 0.48 0.54 0.43 0.54 0.50 0.51 0.49 0.46 0.55 0.60 0.53 0.44 0.51
## [379] 0.52 0.50 0.48 0.52 0.41 0.58 0.50 0.44 0.52 0.42 0.48 0.50 0.49 0.53
## [393] 0.43 0.57 0.44 0.56 0.55 0.46 0.59 0.56 0.56 0.47 0.42 0.47 0.55 0.54
## [407] 0.54 0.48 0.48 0.53 0.55 0.47 0.45 0.47 0.52 0.48 0.49 0.61 0.50 0.55
## [421] 0.51 0.39 0.45 0.51 0.56 0.40 0.61 0.52 0.52 0.53 0.44 0.48 0.46 0.52
## [435] 0.51 0.48 0.56 0.55 0.44 0.54 0.49 0.48 0.46 0.50 0.51 0.46 0.51 0.53
## [449] 0.50 0.47 0.55 0.53 0.53 0.48 0.45 0.53 0.56 0.54 0.48 0.55 0.48 0.50
## [463] 0.52 0.44 0.53 0.39 0.55 0.56 0.52 0.51 0.56 0.43 0.54 0.50 0.53 0.53
## [477] 0.40 0.46 0.46 0.55 0.47 0.50 0.59 0.56 0.45 0.52 0.50 0.57 0.42 0.55
## [491] 0.40 0.42 0.44 0.50 0.53 0.53 0.52 0.54 0.51 0.50 0.41 0.49 0.57 0.60
## [505] 0.52 0.52 0.51 0.55 0.41 0.44 0.37 0.51 0.50 0.54 0.53 0.60 0.54 0.55
## [519] 0.44 0.57 0.47 0.49 0.55 0.47 0.57 0.56 0.54 0.45 0.52 0.53 0.48 0.59
## [533] 0.51 0.44 0.58 0.57 0.56 0.39 0.47 0.47 0.47 0.45 0.45 0.54 0.49 0.48
## [547] 0.45 0.50 0.54 0.45 0.49 0.63 0.56 0.39 0.58 0.43 0.53 0.50 0.53 0.59
## [561] 0.59 0.49 0.56 0.62 0.50 0.46 0.49 0.60 0.52 0.50 0.54 0.50 0.51 0.44
## [575] 0.51 0.48 0.41 0.44 0.41 0.49 0.44 0.39 0.52 0.42 0.54 0.50 0.50 0.44
## [589] 0.53 0.52 0.52 0.52 0.56 0.49 0.54 0.50 0.48 0.43 0.50 0.57 0.53 0.52
## [603] 0.56 0.47 0.50 0.47 0.61 0.44 0.57 0.55 0.52 0.43 0.48 0.55 0.50 0.48
## [617] 0.43 0.48 0.45 0.51 0.58 0.60 0.50 0.55 0.48 0.54 0.59 0.44 0.45 0.52
## [631] 0.46 0.47 0.45 0.46 0.49 0.51 0.59 0.49 0.53 0.49 0.41 0.49 0.54 0.53
## [645] 0.47 0.41 0.45 0.46 0.56 0.56 0.54 0.47 0.38 0.47 0.52 0.49 0.47 0.52
## [659] 0.42 0.51 0.44 0.54 0.45 0.57 0.61 0.48 0.50 0.50 0.49 0.49 0.51 0.52
## [673] 0.62 0.51 0.43 0.54 0.44 0.51 0.60 0.47 0.54 0.41 0.45 0.50 0.41 0.56
## [687] 0.47 0.48 0.51 0.55 0.52 0.43 0.46 0.44 0.57 0.44 0.53 0.53 0.49 0.54
## [701] 0.44 0.45 0.47 0.52 0.51 0.58 0.51 0.52 0.58 0.49 0.53 0.55 0.57 0.54
## [715] 0.51 0.52 0.51 0.51 0.48 0.58 0.53 0.56 0.43 0.54 0.59 0.56 0.56 0.46
## [729] 0.51 0.45 0.52 0.44 0.48 0.46 0.45 0.44 0.53 0.50 0.53 0.55 0.53 0.50
## [743] 0.51 0.51 0.55 0.48 0.52 0.45 0.57 0.46 0.41 0.47 0.53 0.46 0.49 0.56
## [757] 0.50 0.57 0.56 0.55 0.52 0.45 0.53 0.44 0.49 0.57 0.56 0.53 0.46 0.40
## [771] 0.50 0.47 0.48 0.52 0.46 0.52 0.55 0.51 0.51 0.52 0.50 0.48 0.45 0.48
## [785] 0.60 0.53 0.50 0.46 0.45 0.59 0.49 0.41 0.48 0.45 0.51 0.50 0.55 0.58
## [799] 0.41 0.51 0.51 0.55 0.50 0.44 0.45 0.59 0.44 0.55 0.52 0.48 0.49 0.53
## [813] 0.55 0.53 0.51 0.54 0.52 0.51 0.59 0.50 0.49 0.48 0.44 0.50 0.45 0.47
## [827] 0.52 0.42 0.56 0.47 0.47 0.50 0.47 0.49 0.54 0.52 0.50 0.45 0.42 0.47
## [841] 0.48 0.48 0.48 0.52 0.44 0.43 0.55 0.52 0.51 0.46 0.40 0.58 0.49 0.55
## [855] 0.37 0.48 0.47 0.51 0.45 0.47 0.41 0.53 0.45 0.40 0.56 0.51 0.51 0.58
## [869] 0.54 0.51 0.59 0.43 0.39 0.51 0.53 0.43 0.53 0.54 0.49 0.45 0.48 0.42
## [883] 0.48 0.44 0.46 0.56 0.49 0.50 0.52 0.41 0.49 0.49 0.48 0.43 0.56 0.49
## [897] 0.53 0.53 0.58 0.51 0.48 0.55 0.59 0.47 0.55 0.49 0.51 0.52 0.51 0.59
## [911] 0.55 0.53 0.55 0.48 0.58 0.55 0.45 0.47 0.45 0.53 0.45 0.51 0.58 0.55
## [925] 0.53 0.42 0.56 0.52 0.57 0.52 0.49 0.50 0.45 0.55 0.54 0.50 0.43 0.53
## [939] 0.42 0.55 0.48 0.50 0.45 0.43 0.47 0.55 0.49 0.54 0.50 0.46 0.53 0.54
## [953] 0.55 0.40 0.51 0.42 0.55 0.58 0.57 0.50 0.48 0.54 0.45 0.50 0.55 0.52
## [967] 0.49 0.52 0.51 0.49 0.52 0.44 0.54 0.52 0.51 0.53 0.60 0.52 0.51 0.53
## [981] 0.55 0.41 0.51 0.47 0.52 0.57 0.57 0.56 0.45 0.53 0.49 0.51 0.48 0.50
## [995] 0.50 0.45 0.52 0.58 0.51 0.64hist(y)
(lw<-quantile(y,0.025))## 2.5%
## 0.4(up<-quantile(y,0.975))## 97.5%
## 0.6(pvalue<-sum(y>0.7)/length(y))## [1] 0lecture 12
Hypothsis ex-1
library (MASS)
#H0:bwt =3100
View(birthwt)
x<-birthwt$bwt
x_bar<-mean(x)
x_bar## [1] 2944.587n<-length(x)
y<-NULL
for( i in 1:1000){
y[i]<-mean (rnorm(n=n,mean=3100,sd = sd(x)))
}
ci<-quantile(y, c(0.025,0.975))
ci## 2.5% 97.5%
## 2997.545 3202.605# two tailed test
pvalue<-2*mean(y<x_bar)
pvalue## [1] 0.004Hypothsis ex-2(difference)
library(MASS)
#H0: MuS=MuN
#H1: MuS<MuN
x<-birthwt$bwt
x_bar<-mean(x)
(n<-length(x))## [1] 189s_smoke<-birthwt$bwt[birthwt$smoke==1]
s_smoke## [1] 2557 2594 2600 2663 2665 2769 2769 2782 2821 2906 2920 2948 2948 2977 2977
## [16] 2922 3005 3033 3042 3062 3062 3090 3132 3147 3203 3260 3303 3317 3321 3331
## [31] 3374 3430 3444 3572 3629 3637 3643 3651 3651 3756 3856 3884 3940 4238 709
## [46] 1135 1790 1818 1885 1928 1928 1936 2084 2084 2125 2126 2187 2211 2225 2296
## [61] 2296 2353 2367 2381 2381 2410 2410 2414 2424 2466 2466 2466 2495 2495n1<-length(s_smoke)
n1## [1] 74s_non<-birthwt$bwt[birthwt$smoke==0]
n2<-length(s_non)
(x_mean_differ<-mean(s_smoke)-mean(s_non))## [1] -283.7767y_n<-NULL
for(i in 1:1000){
y1<-mean(rnorm(n=n1,mean = x_bar,sd=sd(x)))
y2<-mean(rnorm(n=n2,mean = x_bar,sd=sd(x)))
y_n[i]<-y1-y2
}
(pvalue<-mean(y_n<x_mean_differ))## [1] 0.005y_n<x_mean_differ## [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [121] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [169] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [181] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [193] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [205] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [217] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [229] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [241] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [253] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [265] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [277] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [289] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [301] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [313] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [325] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [337] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [349] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [361] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [373] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [385] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [397] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [409] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [421] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [433] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [445] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [457] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [469] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [481] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [493] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [505] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [517] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [529] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [541] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [553] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [565] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [577] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [589] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [601] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [613] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [625] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [637] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [649] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [661] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [673] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [685] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [697] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [709] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [721] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [733] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [745] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [757] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [769] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [781] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [793] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [805] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [817] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [829] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [841] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [853] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [865] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [877] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [889] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [901] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [913] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [925] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [937] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [949] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [961] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [973] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [985] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [997] FALSE FALSE FALSE FALSElecture 13
Hypothsis ex-1.1
library(MASS)
x<-birthwt$bwt
x_bar<-mean(x)
x_bar## [1] 2944.587n<-length(x)
#set.seed(013)
y<-NULL
for(i in 1: 1000){
y[i]<-mean(rnorm(n = n,mean = 3100,sd = sd(x)))
}
mean(y<x_bar)## [1] 0.003mean(y>x_bar)## [1] 0.997lecture 14
Hypothsis ex-3(difference)
#H0 :yb = ya
#H1 :yb != ya
# paired two tailed test
yb<-c(201,231,221,260,228,237,326,235,240,267)
ya<-c(200,234,216,233,224,216,296,195,207,247)
d<- yb-ya
length(d)## [1] 10mean_d<-mean(d)
sd_d<-sd(d)
m.d<-NULL
for(i in 1:1000){
d_s<-rnorm(10,0,sd_d)
m.d[i]<-mean(d_s)
}
quantile(m.d,0.5)## 50%
## -0.3953587(p<-mean(m.d>mean_d)*2)## [1] 0Assignment
Assignment 01
1
Generate a sample of size 400 from a Poisson distribution with lambda=5. Repeat this process for an adequate time and find the distribution of the mean.
x<-NULL
for(i in 1: 1000){
a<-rpois(n = 400,lambda = 5)
x[i]<-mean(a)
}
hist(x,freq = FALSE ,main = "Histogram of Poisson distribution with lambda=5",xlab="mean",col='7')
curve(dnorm(x,mean(x),sd(x)),add =TRUE,lwd=3,col="red")
According to histogram poisson mean follow normal distribution.
2
Prove the central limit theorem using a binomial distribution bin(10,0.3).
set.seed(34)
x<-rbinom(n = 1000,size = 10,prob = 0.3)
hist(x,main = "Population Distribution",xlab = 'mean',col="ORANGE")
binomial_mean2<-NULL
binomial_mean20<-NULL
binomial_mean70<-NULL
for(i in 1:100){
binomial_mean2[i]<-mean(sample(x = x,size = 2,replace = FALSE))
binomial_mean20[i]<-mean(sample(x = x,size = 20,replace = FALSE))
binomial_mean70[i]<-mean(sample(x = x,size = 70,replace = FALSE))
}
hist(x=binomial_mean2,main = "sample Distribution (n=2)",xlab = 'mean',col="lightgreen")
hist(x=binomial_mean20,main = "sample Distribution (n=20)",xlab = 'mean',col="lightgreen")
hist(x=binomial_mean70,main = "sample Distribution (n=70)",xlab = 'mean',col="lightgreen")
The Central Limit Theorem:The sampling distribution of sample means will
be approximately normal when the sample size is large(n>=30),
irrespective of the distribution of the population
From histogram we show that as n increasing sample mean follows normal distribution which is the main concept of central limit theorem
3
Derive the sampling distribution of sample mean and sample standard deviation (with n and (n-1) in the denominator) when sampling from different population distributions (e.g., exponential(1.5), gamma (5,1), uniform(10,15), Poisson (8) etc.)
exponential(1.5)
set.seed(34)
x<-rexp(n = 400,1.5)
hist(x,main = "Histogram of exponential distribution ",xlab="mean",col="blue")
a<-NULL
b<-NULL
for(i in 1: 1000){
a[i]<-mean(sample(x,50,T))
b[i]<-sd(sample(x,50,T))
}
hist(a,main = "sample distribution ",xlab="mean",col="lightblue")
#By using n-1 denominator standard deviation =
sd(a)## [1] 0.09456174##By using n denominator standard deviation =
sd(a)*sqrt(49/50)## [1] 0.09361135gamma (5,1)
set.seed(34)
x<-rgamma(n = 400,5,1)
hist(x,main = "Histogram of gamma distribution with g~(5,1) ",xlab="mean",col="blue")
a<-NULL
b<-NULL
for(i in 1: 1000){
a[i]<-mean(sample(x,50,T))
b[i]<-sd(sample(x,50,T))
}
hist(a,main = "sample distribution ",xlab="mean",col="lightblue")
#By using n-1 denominator standard deviation =
sd(a)## [1] 0.294428##By using n denominator standard deviation =
sd(a)*sqrt(49/50)## [1] 0.2914688uniform(10,15)
set.seed(34)
x<-runif(n = 400,10,15)
hist(x,main = "Histogram of uniform u~(10,15) distribution ",xlab="mean",col="blue")
a<-NULL
b<-NULL
for(i in 1: 1000){
a[i]<-mean(sample(x,50,T))
b[i]<-sd(sample(x,50,T))
}
hist(a,main = "sample distribution ",xlab="mean",col="lightblue")
#By using n-1 denominator standard deviation =
sd(a)## [1] 0.2169774##By using n denominator standard deviation =
sd(a)*sqrt(49/50)## [1] 0.2147966Poisson (8)
set.seed(34)
x<-rpois(n = 400,lambda = 8)
hist(x,main = "Histogram of Poisson (8) distribution ",xlab="mean",col="blue")
a<-NULL
b<-NULL
for(i in 1: 1000){
a[i]<-mean(sample(x,50,T))
b[i]<-sd(sample(x,50,T))
}
hist(a,main = "sample distribution ",xlab="mean",col="lightblue")
#By using n-1 denominator standard deviation =
sd(a)## [1] 0.4321971##By using n denominator standard deviation =
sd(a)*sqrt(49/50)## [1] 0.42785334
Test the fairness of a coin when the coin follows Bernoulli(0.6) distribution.
#fairness of a coin (H0: coin not fair; H1: coin fair)
set.seed(13)
y<-NULL
for (i in 1:1000){
s<-rbinom(1000,1,0.6)
y[i]<-sum(s)
}
(lw<-quantile(y,0.025))## 2.5%
## 568.975(up<-quantile(y,0.975))## 97.5%
## 631#P(H1)
p_H1<-sum ((568.975<y & y<631)/length(y))
p_H1## [1] 0.949#P(H0)
p_value<-1-p_H1
p_value## [1] 0.0515
Five fair coins are flipped. If the outcomes are assumed independent. Find the probability mass function of the number of heads obtained.
#probability mass function(pmf):
x<-c(0,1,2,3,4,5)
dbinom(x = x,size = 5,prob = .5)## [1] 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125## cdf
pbinom(q = x,size = 5,prob = .5)## [1] 0.03125 0.18750 0.50000 0.81250 0.96875 1.00000plot(x = x,y =dbinom(x = x,size = 5,prob = .5),type="h",xlab = "No of head",ylab = "pmf")
6
It is known that screws produced by a certain company will be defected with probability 0.01 independently of each other. The company sells the screws in package of 10 and offers a money back guarantee that at most 1 of the 10 screws is defective. What proportion of packages sold must the company replace.
#defected,p=.01
#probability of at most 1 of the 10 screws is defective =p(0)+p(1)
x<-dbinom(x = 0,size = 10,prob = .01)
x## [1] 0.9043821y<-dbinom(x = 1,size = 10,prob = .01)
y## [1] 0.09135172#proportion of packages sold must the company replace=
1-x-y## [1] 0.0042662##NOTE:we know that, when n is large but probability p is very small then binomial distribution is similar to poison distribution
### alternative solution
#probability of at most 1 of the 10 screws is defective =p(0)+p(1)
a<-dpois(x = 1,lambda =.1 )
a## [1] 0.09048374b<-dpois(x = 0,lambda =.1 )
b## [1] 0.9048374#proportion of packages sold must the company replace=
1-a-b## [1] 0.004678847
Suppose that the probability that an item produced by certain machine will be defective is 0.1.find the probability that a sample of 10 items will contain at most 1 defective item.
#defected,p=.1
#probability of at most 1 of the 10 screws is defective =p(0)+p(1)
x<-dbinom(x = 0,size = 10,prob = .1)
y<-dbinom(x = 1,size = 10,prob = .1)
x+y## [1] 0.7360989Incourse
1st incourse
Question 1
a
set.seed(99)
A<-rnorm(n = 50,mean = 3,sd = 0.4)
mean(A)## [1] 2.899035sd(A)## [1] 0.4079423b
sampdst01<-replicate(n = 1000, sd(rnorm(3,3,0.4)))
hist(sampdst01)
c
sampdst02<-replicate(10000, sd(rnorm(30,3,0.4)))
hist(sampdst02)
Question 2
a
x<-c(2,4,6,8)
px<-c(.3,.2,.4,.1)
mu<-sum(x*px)
mu## [1] 4.6sigma<-sqrt(sum((x^2)*px)-mu^2)
sigma## [1] 2.009975b
popu<-sample(x,50,replace=T, prob=px)
popu## [1] 6 6 2 8 6 6 2 6 2 2 6 6 6 6 6 4 6 4 4 2 6 6 4 6 2 2 2 6 8 8 6 8 4 4 4 2 2 6
## [39] 8 4 6 4 6 8 6 4 8 4 2 4c
mean(x)## [1] 5all_sam<-combn(x,3)
all_sam## [,1] [,2] [,3] [,4]
## [1,] 2 2 2 4
## [2,] 4 4 6 6
## [3,] 6 8 8 8#MARGIN =2 means column wise equation
#MARGIN =1 means row wise equation
sam_mean<-apply(X = all_sam,MARGIN = 2,mean)
sam_mean## [1] 4.000000 4.666667 5.333333 6.000000mean(sam_mean)## [1] 5