###############################################
#     p_1-p_2: sampling distribution          #
# Repeat 1,000 extractions from a normal      #
############################################### 

sampling.prop.diff <-c()

R = 0 # number of replication
set.seed(123)
repeat{
	
p1<-0.55
p2<-0.75
R=R+1
n1 = n2 = 10 # ...try to change N value to obtain different np and n(1-p) values...

# Random sample from binomial populations, 1 and 2:    
		X1<-rbinom(1,size=n1,prob=p1)
        p1_hat=X1/n1
		X2<-rbinom(1,size=n2,prob=p2)
        p2_hat=X2/n2
        
sampling.prop.diff <-c(sampling.prop.diff,p2_hat-p1_hat)

cat("R=",R,"\n")
if(R==1000)break()
	
	
}
#--------------------------------------------------
# Histogram of sampling distribution:
plot(density(sampling.prop.diff),
     type="l",lwd=2,col="white",bty="n",
     xlim=c(
            (p2-p1)-3*sqrt((p1*(1-p1))/n1+(p2*(1-p2))/n2),
            (p2-p1)+3*sqrt((p1*(1-p1))/n1+(p2*(1-p2))/n2)
            ),
     main=paste("N = ",n1),xlab=expression(hat(p)[2]-hat(p)[1]))
polygon(density(sampling.prop.diff), col = rgb(0.25, 0.99, 0.25, alpha = 0.5),border="white")
abline(v=p2-p1,lwd=1)
#--------------------------------------------------

mean(sampling.prop.diff)
p2-p1
sd(sampling.prop.diff)
sqrt((p1*(1-p1))/n1+(p2*(1-p2))/n2)
#--------------------------------------------------
n1*p1; n1*(1-p1)
n2*p2; n2*(1-p2)
#--------------------------------------------------






###############################################
#     Visualize the two populations           #
#    Normal vs asymmetric chi-square          #
############################################### 

# 1) Normal Population
# ---------------------	
mu1<-12
sd1<-1.5
Population1<-rnorm(50000,mean=mu1,sd=sd1)

# 2) Asimmetric chi-square Population
# ------------------------------------	
mu2=v=8
sd2<-sqrt(2*v)
Population2<-rchisq(50000,df=v)
# ------------------------------------	

# Histogram of population distribution:
plot(density(Population1),
     type="l",lwd=2,col="white",bty="n",
     main="The two populations",xlim=c(0,20),xaxt="n")
axis(1,at=c(0,2,4,6,8,10,12,14,16,18,20))
polygon(density(Population1), col = rgb(0.25, 0.25,0.99, alpha = 0.5),border="white")
abline(v=mu1,lwd=2,col="blue")
polygon(density(Population2), col = rgb(0.99,0.25, 0.25, alpha = 0.5),border="white")
abline(v=mu2,lwd=2,col="red")








###############################################
#     mu1-mu2: sampling distribution          #
# Repeat 1,000 extractions from a normal      #
############################################### 

sampling.mean.diff <-c()

R = 0 # number of replication

# 1) Normal Population
# ---------------------	
mu1<-12
sd1<-1.5

# 2) Asimmetric chi-square Population
# ------------------------------------	
mu2=v=8
sd2<-sqrt(2*v)

set.seed(123)
repeat{

R=R+1
n1 = n2 = 50 # try to change N...

# Random sample from normal population (1)    
		X1<-rnorm(n1,mean=mu1,sd=sd1)
        mu1_hat=sum(X1)/n1
# Random sample from asimmetric population (2)    
		X2<-rchisq(n2,df=v)
        mu2_hat=sum(X2)/n2
        
sampling.mean.diff <-c(sampling.mean.diff,mu1_hat-mu2_hat)

cat("R=",R,"\n")
if(R==1000)break()
	
	
}
#--------------------------------------------------
# Histogram of sampling distribution:
plot(density(sampling.mean.diff),
     type="l",lwd=2,col="white",bty="n",
     main=paste("N = ",n1),xlab=expression(bar(x)[1]-bar(x)[2]))
polygon(density(sampling.mean.diff), col = rgb(0.25, 0.99, 0.25, alpha = 0.5),border="white")
abline(v=mu1-mu2,lwd=1)
#--------------------------------------------------

mean(sampling.mean.diff)
mu1-mu2
sd(sampling.mean.diff)
sqrt(sd1^2/n1+sd2^2/n2)

#--------------------------------------------------









#Reading in the data
cellPhoneReactions <- read.csv(file='https://raw.githubusercontent.com/artofstat/data/master/Chapter10/cell_phone_reaction_times_long.csv')
#To subset data to make the two groups
cell <- subset(cellPhoneReactions, Group =='Cell')
control <- subset(cellPhoneReactions, Group == 'Control')
#To make a hypothesis test comparing the two means
t.test(cell$ReactionTime, control$ReactionTime)
##
## Welch Two Sample t-test
##
## data: cell$ReactionTime and control$ReactionTime
## t = 2.6307, df = 56.696, p-value = 0.01095
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 12.31658 90.87092
## sample estimates:
## mean of x mean of y
## 585.1875 533.5938