Exercise 1.3

library(binom)
x <- seq(0, 100, by=1)
y <- dbinom(0:100, size=100, prob=0.25)
plot(x, y, ylab="P(x)", xlab="n", main = c("X ~ B(100, 0.25)"))




n <- rep(0:100)
probability  <- dbinom(n, 100, 0.25)
chance <- list(n, probability)
df <- as.data.frame(chance, col.names = c("n", "p"))
df
##       n            p
## 1     0 3.207202e-13
## 2     1 1.069067e-11
## 3     2 1.763961e-10
## 4     3 1.920758e-09
## 5     4 1.552613e-08
## 6     5 9.936720e-08
## 7     6 5.244380e-07
## 8     7 2.347484e-06
## 9     8 9.096502e-06
## 10    9 3.099549e-05
## 11   10 9.401965e-05
## 12   11 2.564172e-04
## 13   12 6.339204e-04
## 14   13 1.430384e-03
## 15   14 2.962939e-03
## 16   15 5.662506e-03
## 17   16 1.002735e-02
## 18   17 1.651564e-02
## 19   18 2.538515e-02
## 20   19 3.651899e-02
## 21   20 4.930064e-02
## 22   21 6.260399e-02
## 23   22 7.493508e-02
## 24   23 8.470922e-02
## 25   24 9.059180e-02
## 26   25 9.179969e-02
## 27   26 8.826893e-02
## 28   27 8.064075e-02
## 29   28 7.008066e-02
## 30   29 5.799778e-02
## 31   30 4.575381e-02
## 32   31 3.443835e-02
## 33   32 2.475256e-02
## 34   33 1.700176e-02
## 35   34 1.116782e-02
## 36   35 7.019775e-03
## 37   36 4.224864e-03
## 38   37 2.435958e-03
## 39   38 1.346187e-03
## 40   39 7.133642e-04
## 41   40 3.626268e-04
## 42   41 1.768911e-04
## 43   42 8.282997e-05
## 44   43 3.724138e-05
## 45   44 1.608151e-05
## 46   45 6.670847e-06
## 47   46 2.658671e-06
## 48   47 1.018214e-06
## 49   48 3.747594e-07
## 50   49 1.325680e-07
## 51   50 4.507311e-08
## 52   51 1.472977e-08
## 53   52 4.626660e-09
## 54   53 1.396727e-09
## 55   54 4.052234e-10
## 56   55 1.129714e-10
## 57   56 3.026019e-11
## 58   57 7.786248e-12
## 59   58 1.924188e-12
## 60   59 4.565869e-13
## 61   60 1.040003e-13
## 62   61 2.273232e-14
## 63   62 4.766453e-15
## 64   63 9.583346e-16
## 65   64 1.846791e-16
## 66   65 3.409459e-17
## 67   66 6.026822e-18
## 68   67 1.019462e-18
## 69   68 1.649130e-19
## 70   69 2.549380e-20
## 71   70 3.763371e-21
## 72   71 5.300523e-22
## 73   72 7.116443e-23
## 74   73 9.098648e-24
## 75   74 1.106592e-24
## 76   75 1.278729e-25
## 77   76 1.402115e-26
## 78   77 1.456743e-27
## 79   78 1.431841e-28
## 80   79 1.329135e-29
## 81   80 1.162993e-30
## 82   81 9.571963e-32
## 83   82 7.392979e-33
## 84   83 5.344322e-34
## 85   84 3.605297e-35
## 86   85 2.262147e-36
## 87   86 1.315202e-37
## 88   87 7.054722e-39
## 89   88 3.473916e-40
## 90   89 1.561311e-41
## 91   90 6.360895e-43
## 92   91 2.329998e-44
## 93   92 7.597820e-46
## 94   93 2.178587e-47
## 95   94 5.407839e-49
## 96   95 1.138492e-50
## 97   96 1.976549e-52
## 98   97 2.716906e-54
## 99   98 2.772353e-56
## 100  99 1.866905e-58
## 101 100 6.223015e-61
n <- 100
p <- 0.25
q <- 1-p

mean <- n*p
variance <- n*p*q
std <- sqrt(variance)
mean
## [1] 25
variance
## [1] 18.75
std 
## [1] 4.330127
data <- 50
z_score <- (data-mean)/std 
z_score 
## [1] 5.773503
options("scipen" = 999)
p_value <- pnorm(q=z_score, lower.tail=FALSE)
p_value
## [1] 0.000000003882018
p_value <- pnorm(q=-5, lower.tail=TRUE)
p_value
## [1] 0.0000002866516



Exercise 1.12

binom.confint(0, 25, conf.level=0.95, method="asymptotic")
##       method x  n mean lower upper
## 1 asymptotic 0 25    0     0     0
binom.confint(0, 25, conf.level=0.95, method="wilson")
##   method x  n mean lower     upper
## 1 wilson 0 25    0     0 0.1331923



Reference

[1] Alan Agresti, 『범주형 자료분석 개론』, 박태성, 이승연 역, 자유아카데미, 2020
[2] Alan Agresti, 『Categorical Data Analysis (3rd Edition)』, Wiley, 2018