### Today we continue our exploration of the t-distributioin. Recall for normal random variables the student-t distribution does a great job allowing us to use s.  Assuming that the null is H0: mu=15,  we graph a histogram of the  t statistic  after N trials of our n=4 Gosset experiment when the null is true.  In red is the student t distribution in while in blue we see the normal. 

mu=15
sigma=3
n=4
N=10000
M=n*N
df=n-1

X=rnorm(M,mu,sigma)
dim(X)<-c(N,n)
Xbar=apply(X,1,mean)
s=apply(X,1,sd)
T=(Xbar-mu)/(s/sqrt(n))
#ST=sort(T)
CutOff=8
hist(T[abs(T)<CutOff],freq=FALSE,nclass=floor(sqrt(N/2)),ylim=c(0,dnorm(0,0,1)),main=paste("Histogram of t-statistic simulation, df=",df),xlab="s-standard deviations from mean",col="yellow")
X=seq(-10,10,by=0.01)
points(X,dnorm(X,0,1),type="l",col="blue",lwd=2)
points(X,dt(X,df),type="l",col="red",lwd=2)



#########

# Differecne of means. Here we imagine that Gosset wished to compare two batches.  Perhaps the morning and afternoon Guiness Crew's output. All else is the same and he'd like to decide whether the mean ick really in the same (See RMay15.txt).

# We will let Xbar12 denote the difference in means.  We will first look at the 90 percent confidence interval and see if it contains zero.   To simulate this require making a choice, here we assume the Null is true. 

ErrorRate=0.05
Conf=1-2*Error
mu=15
sigma=3
n1=4
n2=4

X1=rnorm(n1,mu,sigma)
Xbar1=mean(X1)
s1=sd(X1)
SE1=s1/sqrt(n1)
df1=n1-1
tstar1=abs(qt(ErrorRate,df1))
CInt1=c(Xbar1-tstar1*SE1,Xbar1,Xbar1+tstar1*SE1)

X2=rnorm(n2,mu,sigma)
Xbar2=mean(X2)
s2=sd(X2)
SE2=s2/sqrt(n2)
df2=n2-1
tstar2=abs(qt(ErrorRate,df2))
tstar2
CInt2=c(Xbar2-tstar2*SE2,Xbar2,Xbar2+tstar2*SE2)

Xbar12=Xbar1-Xbar2
SE12=sqrt(s1^2/n1+s2^2/n2)
df12=((s1^2/n1+s2^2/n2)^2)/((1/(n1-1))*(s1^2/n1)^2+(1/(n2-1))*(s2^2/n2)^2)
df12=((SE12)^4)/((SE1^4/(n1-1))+(SE2^4/(n2-1)))
tstar12=abs(qt(ErrorRate,df12))
CInt12=c(Xbar12-tstar12*SE12,Xbar12,Xbar12+tstar12*SE12)

#Here is the confidence interval with our approximation of the difference from in the data in the middle. 

CInt12


####### Rules of thumb


 # In practice random variables are only approximately normal.  Here we compare the normal, the t-distribution, and a sample in some cases that satisfy our book's rules of thumb.   Here we test that between 15 and 50 samples a we can apply out t-test to a mildly skewed distribution, like: 


X1=seq(-10,10,by=.005)
Y1=dgamma((X1+5),5,1)
plot(X1,Y1,type="l",main="A moderately skewed distribution",xlab="Values",ylab="Probability Density")

R=rgamma(20,5,1)-5
R
hist(R,nclass=3)

mu=5*1
n=20
N=10000
M=n*N
df=n-1

X=rgamma(M,5,1)
dim(X)<-c(N,n)
Xbar=apply(X,1,mean)
s=apply(X,1,sd)
T=(Xbar-mu)/(s/sqrt(n))
#ST=sort(T)
CutOff=8
hist(T[abs(T)<CutOff],freq=FALSE,nclass=floor(sqrt(N/2)),ylim=c(0,dnorm(0,0,1)),main=paste("Using the t-test for a moderately skewed, df=",df),xlab="s-standard deviations from mean",col="yellow")
X=seq(-10,10,by=0.01)
points(X,dnorm(X,0,1),type="l",col="blue",lwd=2)
points(X,dt(X,df),type="l",col="red",lwd=2)

#  For our moderately skewed random variable is the difference between z and t appear as significant as the difference between the true distribution and either of them?

#Conclusion:  By the time CLT really kicks in t and z will be basically indistinguishable.  



####

# Here we compare the normal, the t-distribution, and a sample in some cases that satisfy our book's rules of thumb.   Here we test that with more than 50 samples we can apply out t-test to a skewed distribution, like:

X1=seq(0,6,by=.01)
Y1=dexp(X1,1)
plot(X1,Y1,type="l",main="A skewed distribution",xlab="Values",ylab="Probability Density")

mu=1
n=50
N=10000
M=n*N
df=n-1

X=rexp(M,1)
dim(X)<-c(N,n)
Xbar=apply(X,1,mean)
s=apply(X,1,sd)
T=(Xbar-mu)/(s/sqrt(n))
#ST=sort(T)
CutOff=8
hist(T[abs(T)<CutOff],freq=FALSE,nclass=floor(sqrt(N/2)),ylim=c(0,dnorm(0,0,1)),main=paste("Using the t-test for a moderately skewed, df=",df),xlab="s-standard deviations from mean",col="yellow")
X0=seq(-10,10,by=0.01)
points(X0,dnorm(X0,0,1),type="l",col="blue",lwd=2)
points(X0,dt(X0,df),type="l",col="red",lwd=2)


#  For our   skewed random variable is the difference between z and t appear as significant as the difference between the true distribution and either of them?

#Conclusion:  By the time CLT really kicks in t and z will be basically indistinguishable.