#Example 3
# Testing hypothesis about the population proportion 

#assumed population proportion
pi0 <- 0.20  
#sample size
n <- 200
# number of families in the sample living below
# the poverty level
x <- 45
#sample proportion 
p <- x/n
p

#hypotheses
#H0: pi = 0.20
#H1: pi not= 0.20

#test statistic
ts <- (p - pi0)/sqrt(pi0*(1-pi0)/n)
ts      

#lower critical value - 2.5th percentile (0.025th quantile)
lower_z025 <- qnorm(.025)
lower_z025

#upper critical value - 97.5th percentile (0.975th quantile)
upper_z025 <- qnorm(.975)
upper_z025

#plot of standard normal density function
curve(dnorm(x), from = -4, to = 4,
      main = 'Standard Normal Distribution',
      ylab = 'Density', xlab = '')

#location of the test statistic and critical values
abline(v=ts,col="blue")
#location of the lower critical value
abline(v=lower_z025,col="red")
#location of the upper critical value
abline(v=upper_z025,col="red")



#determination of the p-value

#plot of standard normal density function
curve(dnorm(x), from = -4, to = 4,
      main = 'Standard Normal Distribution',
      ylab = 'Density', xlab = '')

#location of the negative and postive absolute values of 
#test statistic with hosizonal line showing the value of
#the density function
abline(v=abs(ts),col="blue")
abline(v=-abs(ts),col="blue")

#value of the density function when the quantile is the 
#postive or negative of absolute value of the test statistic
dnorm(abs(ts))
dnorm(-abs(ts))

abline(h=dnorm(abs(ts)),col="red")


#P(Z > abs(ts))
pv_up <- pnorm(abs(ts),lower.tail=FALSE)
pv_up
#P(Z < -abs(ts))
pv_lo <-pnorm(-abs(ts),lower.tail=TRUE) 
pv_lo
pvalue <-pv_up + pv_lo
pvalue