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

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

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

Var(H)≤ 
  unanswered  
 
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:  
 unanswered  
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:  
 unanswered

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.

Normal Table
Show

[H−1.96⋅3−−−−−−√n−−√,H+1.96⋅3−−−−−−√n−−√] 
 [H−1.963n−−√,H+1.963n−−√] 
 [H−1.96⋅3–√n−−√,H+1.96⋅3–√n−−√] 
 [H−1.96⋅3n−−√,H+1.96⋅3n−−√] 
unanswered