Introduction
This page accompanies Section 7.3 in DeGroot and Schervish. When the data are IID normal, then the sample mean and sample variance are independent; the former has a normal distribution; and the latter is proportional to a chi-square random variable with n − 1 degrees of freedom. See Theorem 7.3.1 in DeGroot and Schervish.
We will suppose we have n = 30, μ = 10, and σ = 5.
Distribution of Sample Mean
Direct Look-Up
What is the probability that the sample mean is between 8 and 12?
The pnorm function
on-line
help computes the normal distribution function.
Inverse Look-Up
What are two numbers such that the probability that the sample mean is between those two numbers is 0.95 and the two numbers are of the form μ ± c?
The qnorm function
on-line
help computes the normal inverse distribution function.
Distribution of Sample Variance
Direct Look-Up
What is the probability that the sample variance is between 9 and 49?
The pchisq function
on-line
help computes the chi-square distribution function.
Inverse Look-Up
What are two numbers such that the probability that the sample variance is between those two numbers is 0.95 and the probability that the sample variance is below the interval is 0.025 and the probability it is above the interval is also 0.025?
The qchisq function
on-line
help computes the chi-square inverse distribution function.
