sampling_distribution

#1

pop.mean<-6
pop.sd<-2
x<-rnorm(1000000,pop.mean,pop.sd)
mean(x)
sd(x)

n<-4
# how do i draw a sample of 4 from x ?
?sample
sample(x,size=n)
mean(sample(x, size=n))
rep_times<-1000
x_bar<-replicate(rep_times, mean(sample(x, size=n)))
hist(x_bar,breaks=100)
mean(x_bar)
sd(x_bar)

qqnorm(x_bar)
qqline(x_bar, col=2)
hist(x_bar, breaks = 100, prob=T)
curve(dnorm(x, mean=6, sd=1), col='blue', add=T)


#2
n=4
# we use the replication f
samples<-replicate(rep_times, sample(x, size=n))
sample_mean<-apply(samples, 2, mean)
sample_sd<-apply(samples,2, sd)
sample_variance<-apply(samples,2,var)

cor(sample_mean, sample_sd^2)
cor(sample_mean, sample_variance)

rescaled_sample_vs<-(n-1)*sample_sd^2/pop.sd^2
hist(rescaled_sample_vs, breaks=100, prob=T)
curve(dnorm(x, mean=mean(rescaled_sample_vs),
            sd=sd(rescaled_sample_vs)),col='blue',add=T)
curve(dchisq(x, df=n-1),col='red',add=T)


#3
samples2<-replicate(rep_times, sample(x, size=n))
sample2_mean<-apply(samples2, 2, mean)
sample2_sd<-apply(samples2,2, sd)

sample_t<-(sample2_mean-pop.mean)/(sample2_sd/sqrt(n))
hist(sample_t, breaks=100, prob=T)
curve(dnorm(x, mean=mean(sample_t),
            sd=sd(sample_t)),col='blue',add=T)
curve(dt(x, df=n-1),col='red',add=T)