#Fat content (in percentage) of $n=10$ randomly selected hot dogs
x = c(25.2,21.3,22.8,17,29.8,21,25.5,16,20.9,19.5)
qqnorm(x)
qqline(x)
n = length(x)
s = sd(x)
alpha=0.05
mean(x)+c(-1,1)*qt(1-alpha/2,n-1)*s/sqrt(n)

#Direct use of routine in R
t.test(x,conf.level=0.95)$conf.int

#Prediction interval:

mean(x)+c(-1,1)*qt(1-alpha/2,n-1)*s*sqrt(1+1/n)

#Confidence interval for sigma^2

(n-1)*s^2*1/qchisq(c(1-alpha/2,alpha/2),n-1)

library(asbio)
ci.sigma(x)

#Confidence interval for sigma
sqrt(n-1)*s*1/sqrt(qchisq(c(1-alpha/2,alpha/2),n-1))

sqrt(ci.sigma(x)$ci)