#Biostatistics 100

#ANALOGOUS FUNCTIONS FOR 
#NORMAL AND T DISTRIBUTIONS

#NORMAL DISTRIBUTION
mu=0        #parameter for mean
sigma=1    #paramater for standard deviation
n=1000     #number of randomly generated data points
X=1.96      #quantile X
P=0.95      # cumulative probability phi(X)
rnorm(n,mu,sigma)   #to generate random data points
dnorm(X,mu,sigma)   #P(X) from X
pnorm(X,mu,sigma)   #phi(X) from X
qnorm(P,mu,sigma)   #X from phi(X)

#t DISTRIBUTION
df=5       #degrees of freedom parameter
n=1000   #number of randomly generated data points
X=1.96    #quantile X
P=0.95   # cumulative probability phi(X) 
rt(n,df)  #to generate random data points
dt(X,df)  #P(X) from X
pt(X,df)  #phi(X) from X
qt(P,df)   #X from phi(X)

#CONFIDENCE INTERVALS
mu=50
sigma=10
n=100

X=rnorm(100,mu,sigma)
alpha=0.05   

#NORMAL DISTRIBUTION
L=qnorm((alpha/2),0,1)
L
U=qnorm((1-alpha/2),0,1)
U
#confidence interval:
mu+L*(sigma/sqrt(n))
mu+U*(sigma/sqrt(n))

#t DISTRIBUTION
df=n-1
s=sqrt(var(X))
L=qt((alpha/2),df)
L
U=qt((1-alpha/2),df)
U
#confidence interval:
mu+L*(s/sqrt(n))
mu+U*(s/sqrt(n))
#NOTE: These values don't match MathCad
#because they are based on a different sample!