Let X be a continuous random variable. We know that it takes values between 0 and 3 , but we do not know its distribution or its mean and variance. We are interested in estimating the mean of X, which we denote by h . We will use as a conservative value (upper bound) for the standard deviation of X . To estimate , we take i.i.d. samples , which all have the same distribution as X , and compute the sample mean .

Question

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

2. Given the available information, the smallest upper bound for var(H) is:

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

This minimum value of n is:

4. We would like to be at least 99% sure that our estimate is within 0.05 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:

5. Assume that X is uniformly distributed on [0,3] . Using the Central Limit Theorem, identify the most appropriate expression for a 95% confidence interval for h .

Answer 1

1. H

Answer 2

5. H-1.96*(3^0.5)/((4*n)^0.5),
H+1.96*(3^0.5)/((4*n)^0.5)

Answer 3

1.1. h

1.2. 2.25/n

2. 22500

3. 90000