Exercise: variance of the sample mean
#Objective It’s a short one today - we’ll be talking about the central limit theorem soon and I want to make sure the students are convinced that scaling by $\sqrt{n}$ stabilizes the variance of the mean. Everything else is for review.
Exercise
##Overview Yesterday you sampled some data relating to the profit (in cents) per gallon of gas at gas stations nationally, and used the data to estimate the expected profit per gallon. Today we follow up with some discussion questions.
Calculations
I took a sample and my numbers were
- This sample has mean 1.37.
- The problem told us that the population variance is $\sigma^2 = 9$
- The formula for calculating sample mean is ${\bar X} = \frac{1}{n} \sum_{i=1}^n X_i$.
- We can use the rules of expectation to find $E ({\bar X}) = \frac{1}{n} \sum_{i=1}^n E(X_i)$
- Use the rules of variance to find ${\rm var} ({\bar X}) = \frac{1}{n^2} \sum_{i=1}^n {\rm var} (X_i) = \frac{1}{n^2} \times n \sigma^2 = \frac{\sigma^2}{n}$.
- Here’s the Q-Q plot of my data (on back)
Discussion
- What constant that depends on n (call it $c_n$) can you multiply by ${\bar X}$ so that the variance ${\rm var} (c_n {\bar X})$ doesn’t change with sample size?
- Hint: use the properties of variance