Question

Let X be a continuous random variable. We know that it takes values between 0 and 6 , but we do not know its distribution or its mean and variance, although we know that its variance is at most 4 . We are interested in estimating the mean of X , which we denote by h . To estimate h , we take n i.i.d. samples X1,…,Xn , which all have the same distribution as X , and compute the sample mean.

H=1n∑i=1nXi.

1. Express your answers for this part in terms of h and n using standard notation.

a) E[H]=


Given the available information, the smallest upper bound for Var(H) that we can assert/guarantee is:

b) Var(H)≤

2. Calculate the smallest possible value of n such that the standard deviation of H is guaranteed to be at most 0.01.

This minimum value of n is:

3. We would like to be at least 96% sure that our estimate is within 0.02 of the true mean h. Using the Chebyshev inequality, calculate the minimum value of n that will achieve this.

This minimum value of n is:

4. Suppose now that X is uniformly distributed on [h−3,h+3], for some unknown h. Using the Central Limit Theorem, identify the most appropriate expression for a 95% confidence interval for h. You may want to refer to the normal table