The sample mean
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= 1/n ∑i=1 to n Xi.
1) Express your answers for this part in terms of h and n using standard notation.
a) E[H]=
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Given the available information, the smallest upper bound for Var(H) that we can assert/guarantee is:
b) Var(H)≤
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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:
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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:
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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.
a) [H − (√1.96⋅3/√n),H + (√1.96⋅3/√n)]
b) [H− (1.96/√3n),H + (1.96/√3n)]
c) [H− (1.96⋅√3/√n),H + (1.96⋅√3/√n)]
d) [H− (1.96⋅3/ √n),H + (1.96⋅3/√n)]
3 answers