Normal Distribution
?rnorm()
## starting httpd help server ... done
rnorm(100, mean = 0, sd = 1)
## [1] 0.213197260 -0.946108377 -0.032133801 0.722782905 0.381404975
## [6] -2.519191628 -1.111409720 1.455713255 0.162745476 -1.942185455
## [11] -1.513174438 1.019906490 -0.005654242 0.367237927 0.470619466
## [16] 0.332851175 -0.745356491 -0.975314081 -0.339830233 1.155817394
## [21] -1.462386294 2.242478350 0.431883842 -0.618532987 -0.885694814
## [26] -1.050886036 -0.376589668 0.336525967 -1.401716799 0.364662446
## [31] -0.848841787 1.221889401 -0.240583815 0.674600334 -1.387383525
## [36] -1.108229251 -0.714976291 0.153054000 1.376335983 -1.293615906
## [41] -0.904619654 0.877271410 1.574654625 0.275771329 0.825699056
## [46] -0.139870842 -0.078631767 -1.280820702 -0.331295559 0.026099570
## [51] 0.335708854 -0.342467371 0.845364338 -0.691286591 -0.433963124
## [56] 0.622734553 0.592378120 1.861478201 -0.453190640 -0.364706636
## [61] -1.626814147 1.004376352 -0.884022191 -1.302239673 -1.365518546
## [66] -1.008864985 -0.099234075 -0.164381362 -1.426353328 -0.552789459
## [71] -0.630338274 -0.226421658 -0.585671684 0.336165788 -1.391497128
## [76] -1.718842011 0.319832386 0.133656941 -1.416532084 -0.902670330
## [81] -1.067722022 0.401877369 -0.858543950 0.339550709 -1.325981385
## [86] 0.062448630 -1.517878192 -1.466480140 0.135854554 0.363545782
## [91] -0.695062553 -0.533318082 -0.391628192 0.346701035 -1.476327768
## [96] 1.454486242 0.626987321 0.915074854 -1.802122796 0.002510583
hist(rnorm(100, mean = 0, sd = 1))
hist(rnorm(10000, mean = 0, sd = 1))
hist(rnorm(10000, mean = 170, sd = 10))
dnorm(0, mean = 0, sd = 1)
## [1] 0.3989423
dnorm(170, mean = 170, sd = 10)
## [1] 0.03989423
curve(dnorm, -3, 3)
curve(pnorm, -3, 3)
pnorm(1)
## [1] 0.8413447
pnorm(180, mean = 170, sd = 10)
## [1] 0.8413447
pnorm(160, mean = 170, sd = 10)
## [1] 0.1586553
pnorm(180, mean = 170, sd = 10) - pnorm(160, mean = 170, sd = 10)
## [1] 0.6826895
pnorm(190, mean = 170, sd = 10) - pnorm(150, mean = 170, sd = 10)
## [1] 0.9544997
pnorm(200, mean = 170, sd = 10) - pnorm(140, mean = 170, sd = 10)
## [1] 0.9973002
barplot(table(sample(1:6, size = 100000, replace=TRUE)))
hist(runif(1000,0,5))
hist(rpois(5000, lambda = 3))
curl::curl_download('https://raw.githubusercontent.com/ywchiu/fda_course/main/cdc.Rdata', 'cdc.Rdata')
load("cdc.Rdata")
hist(cdc$weight)
shapiro.test(rnorm(5000) )
##
## Shapiro-Wilk normality test
##
## data: rnorm(5000)
## W = 0.99971, p-value = 0.7409
shapiro.test(sample(cdc$weight,5000) )
##
## Shapiro-Wilk normality test
##
## data: sample(cdc$weight, 5000)
## W = 0.96329, p-value < 2.2e-16
Sample Distribution
hist(cdc$height)
sample_means <- rep(NA, 3)
sample_means
## [1] NA NA NA
sample_means10 <- rep(NA, 5000)
sample_means50 <- rep(NA, 5000)
sample_means100 <- rep(NA, 5000)
for (i in 1:5000){
sample10 <- sample(cdc$height, 10)
sample_means10[i]<-mean(sample10)
sample50 <- sample(cdc$height, 50)
sample_means50[i]<-mean(sample50)
sample100 <- sample(cdc$height, 100)
sample_means100[i]<-mean(sample100)
}
par(mfrow =c(3,1))
xlimits <- range(sample_means10)
hist(sample_means10, breaks = 20, xlim = xlimits)
hist(sample_means50, breaks = 20, xlim = xlimits)
hist(sample_means100, breaks = 20, xlim = xlimits)
mean(cdc$height)
## [1] 67.1829
population <- cdc$height
set.seed(123)
sample(1:42, 6)
## [1] 31 15 14 3 37 40
set.seed(123)
samp <- sample(population, 50)
mean(samp)
## [1] 67.4
hist(samp)
pnorm(1) - pnorm(-1)
## [1] 0.6826895
pnorm(2) - pnorm(-2)
## [1] 0.9544997
pnorm(1.96) - pnorm(-1.96)
## [1] 0.9500042
mean(samp)
## [1] 67.4
sde <- sd(samp) / sqrt(50)
sde * 1.96
## [1] 1.12
upper <- mean(samp) + 1.96 * sde
lower <- mean(samp) - 1.96 * sde
print(mean(cdc$height))
## [1] 67.1829
print(lower)
## [1] 66.28
print(upper)
## [1] 68.52
print(upper - lower)
## [1] 2.24
# sample 100
set.seed(123)
samp <- sample(population, 100)
sde <- sd(samp) / sqrt(100)
upper <- mean(samp) + 1.96 * sde
lower <- mean(samp) - 1.96 * sde
print(mean(cdc$height))
## [1] 67.1829
print(lower)
## [1] 66.66719
print(upper)
## [1] 68.27281
print(upper - lower)
## [1] 1.605615
# sample 200
set.seed(123)
samp <- sample(population, 200)
sde <- sd(samp) / sqrt(200)
upper <- mean(samp) + 1.96 * sde
lower <- mean(samp) - 1.96 * sde
print(mean(cdc$height))
## [1] 67.1829
print(lower)
## [1] 66.83475
print(upper)
## [1] 67.98525
print(upper - lower)
## [1] 1.150506
qnorm(0.995)
## [1] 2.575829
pnorm(2.58) - pnorm(-2.58)
## [1] 0.99012
# 95% 信心水準
set.seed(123)
samp <- sample(population, 100)
sde <- sd(samp) / sqrt(100)
upper <- mean(samp) + 1.96 * sde
lower <- mean(samp) - 1.96 * sde
print(mean(cdc$height))
## [1] 67.1829
print(lower)
## [1] 66.66719
print(upper)
## [1] 68.27281
print(upper - lower)
## [1] 1.605615
# 99% 信心水準
set.seed(123)
samp <- sample(population, 100)
sde <- sd(samp) / sqrt(100)
upper <- mean(samp) + 2.58 * sde
lower <- mean(samp) - 2.58 * sde
print(mean(cdc$height))
## [1] 67.1829
print(lower)
## [1] 66.41324
print(upper)
## [1] 68.52676
print(upper - lower)
## [1] 2.113514
