# Unit 2 for R
# 030515 prepared by C. Andy Tsao
# Using R for finding CI and perform hypotheses testing
# for one normal popopulation (with variance known/unknown)

# We will use the SimpleR package (Joh Verzani)
# It is highly recommend you have his SimpleR.pdf handy
# You can refer to the chapters of confidence intervals (p59-65),
# hypotheses testing (p66-68).

# Install (first time only) then load "Simple"

# Input data
# x<-c(36,37,34,38,39,36)

theta<-36.5;sigma<-1;
x<-rnorm(6,theta,sigma)
hist(x)

xbar<-mean(x);s<-sd(x);n<-length(x);

# For testing H_0: \theta \leq theta0 vs H_1: \theta > \theta0

theta0<- 36; sigma<-1;

z<-(xbar-theta0)/(sigma/sqrt(n))
t<-(xbar-theta0)/(s/sqrt(n))

pvz<-1-pnorm(z)
pvt<-1-pt(t,n-1)

c(xbar,sigma,s,pvz,pvt)

# For testing H_0: \theta \geq theta0 vs H_1: \theta < \theta0
theta0<- 36; sigma<-1;

z<-(xbar-theta0)/(sigma/sqrt(n))
t<-(xbar-theta0)/(s/sqrt(n))
pvz<-pnorm(z);
pvt<-pt(t,n-1);
c(xbar,sigma,s,pvz,pvt)

# For testing H_0: \theta = theta0 vs H_1: \theta \neq \theta0
theta0<- 36; sigma<-1;

z<-(xbar-theta0)/(sigma/sqrt(n))
t<-(xbar-theta0)/(s/sqrt(n))
pvz<-2*pnorm(-abs(z));
pvt<-2*pt(-abs(t),n-1);
c(xbar,sigma,s,pvz,pvt)

simple.z.test(x,sigma,0.9)
simple.z.test(x,sigma,0.95)
t.test(x,alternative = c("two.sided"),mu = theta0,conf.level = 0.95)
t.test(x,alternative = c("greater"),mu = theta0,conf.level = 0.9)