#Example 1
# Testing hypothesis about the population mean when the
# population standard deviation is known


#assumed population mean
mu <- 15
#sample size
n <- 70
#sample mean 
xbar <- 14.18
# population standard deviation
sigma <- 3

#hypotheses
#H0: mu >= 15
#H1: mu < 15



#test statistic
ts <- (xbar - mu)/(sigma/sqrt(n))
ts      

#lower critical value - 5th percentile (0.05th quantile)
lower_z05 <- qnorm(.05)
lower_z05

#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 value
abline(v=ts,col="blue")
#location of the lower critical value
abline(v=lower_z05,col="red")


#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 test statistic 
abline(v=ts,col="blue")

#p-value
#P(Z < ts)
pvalue <-pnorm(ts,lower.tail = TRUE)
pvalue