While exploring some ancient ruins, you discover a black box that outputs a random number every minute. All that you can discern from the writing on the box is that the numbers are all drawn independently from a distribution with variance 2.

You want to estimate the mean of the distribution, but don't want to wait any longer than necessary to collect samples. What is the minimum number of samples that you need to collect for you to be at least 95% sure that your sample average is within 0.1 of the true value? (Do not use the CLT, as this will give an approximation and not a guarantee.)

Hint: If you are not sure how to start, suppose you only took four samples, X1,X2,X3 and X4. What is the best upper bound you can nd on the probability that Xn=41(X1+X2+X3+X4) differs from by at most 0.1? Is four samples enough?