#---------------------------------------------------------------------#
#EXAMPLE 1
# Confidence interval estimate of the population mean 
# when the population standard deviation is known

#sample size
n <- 60
#sample mean
xbar <- 6.5
#population standard deviation
sigma <- 4
#97.5th percentile of standard normal distribution
z025 <- qnorm(.975)
z025
#standard error
se <- sigma/sqrt(n)
se
#Lower confidence limit
LCL <- xbar - (z025 * se)
LCL
#Upper confidence limit
UCL <- xbar + (z025 * se)
UCL
#95% confidence interval estimate of mean
cbind(LCL, UCL)


#----------------------------------------------------------------------#
#EXAMPLE 2
# Confidence interval estimate of the population mean 
# when the population standard deviation is unknown

#sample size
n <- 8
#create vector of numbers
x <- c(3.5, 4.2, 5.3, 4.7, 3.4, 4.6, 4.9, 3.7)
#sample mean
xbar <- mean(x)
#sample standard deviation
s <- sd(x)
#97.5th percentile of the t-distribution with 7 degrees of freedom
t025_7 <- qt(.975,7)
t025_7
#standard error
se <-  (s/sqrt(n))
se
#Lower confidence limit
LCL <- xbar - (t025_7 * se)
LCL
#Upper confidence limit
UCL <- xbar + (t025_7 * se)
UCL
#95% confidence interval estimate of mean
cbind(LCL, UCL)


#-----------------------------------------------------------------------#
#EXAMPLE 3
# Confidence interval estimate of the population proportion

#sample size
n <- 400
# number of defective component
x <- 60
#sample proportion
p <- x/n
p
#99.5th percentile of the standard normal distribution
z005 <- qnorm(.995)
z005
#standard error
se <- sqrt(p*(1-p)/n)
se
#Lower confidence limit
LCL <- p - (z005 *se)
LCL
#Upper confidence limit
UCL <- p + (z005 *se)
UCL
#99% confidence interval estimate of proportion
cbind(LCL, UCL)


#-----------------------------------------------------------------------#
#EXAMPLE 4
# Confidence interval estimate of the population variance 

#sample size
n <- 10
#create vector of numbers
x <- c(20.5, 19.8, 21.1, 20.2, 18.9, 19.6, 20.7, 20.1, 19.8, 19.0)
#sample standard deviation
s <- sd(x)
s
#95th percentile of the chi-square distribution with 9 degrees of freedom
chi95_9 <- qchisq(.95,9)
chi95_9
#5th percentile of the chi-square distribution with 9 degrees of freedom
chi5_9 <- qchisq(.05,9)
chi5_9
#Lower confidence limit for variance
LCL <- ((n-1)*s^2)/chi95_9
LCL
#Upper confidence limit for variance
UCL <- ((n-1)*s^2)/chi5_9
UCL
#90% confidence limit estimate of variance
cbind(LCL,UCL)
#90% confidence limit estimate of standard deviation
cbind(sqrt(LCL), sqrt(UCL))

#-----------------------------------------------------------------------#