Question

Consider a standard uniform density. The mean for this density is .5 and the variance is 1 / 12. You sample 1,000 observations from this distribution and take the sample mean, what value would you expect it to be near? Give your answer to at least one decimal place.

Answer 1

Can you build a confidence interval?

Variance of the population is 1/12
Standard Deviation of the population is 1/sqrt(12)
Standard Deviation of the Mean is 1/sqrt(12000)

Is the Sample Mean a biased estimator or not?

Answer 2

This is a guess on my part. You're given some random variable \(X\sim \text{Unif}(0,1)\), and it looks like you have to find \(E(\bar{X})\).
\[\bar{X}=\frac{1}{n}\sum_{i=1}^nX_i~~\Rightarrow~~E(\bar{X})=E\left(\frac{1}{n}\sum_{i=1}^nX_i\right)=\frac{1}{n}\sum_{i=1}^nE(X_i)=E(X_i)\]
I'm not sure if the 1000 observations is a red herring or not...