# Confidence interval estimate and hypothesis testing for
#of the difference between two population proportions

#Classroom Lecture
n1 <- 150
pass1 <- 107
fail1 <- 43
#sample proportion
p1 <- pass1/n1
p1
#Audio-Visual Instruction
n2 <- 100
pass2 <- 63
fail2 <- 37
#sample proportion
p2 <- pass2/n2
p2
#difference bertween sample proportions
p1-p2
#standard error
se <- sqrt((p1*(1-p1)/n1) +(p1*(1-p1)/n1))
se

#97.5th percentile of the standard normal distribution
z025 <- qnorm(.975)
z025

#Lower confidence limit
LCL <- (p1-p2) - (z025 *se)
LCL

#Upper confidence limit
UCL <- (p1-p2) + (z025 *se)
UCL

#95% confidence interval estimate of pi1-pi2
cbind(LCL, UCL)


#H0: pi1-pi2 = 0; H1: pi1-pi2 > 0
#pooled proportion
pbar <- (n1/(n1+n2)*p1) + (n2/(n1+n2)*p2) 
pbar
sep <- sqrt(pbar*(1-pbar)*(1/n1 + 1/n2))
sep
#Compute test statistic
ts <- (p1-p2)/sep
ts
#p-value = P(Z > ts)
pvalue <-pnorm(ts,lower.tail = FALSE )
pvalue