QCEVault

Statistical inference — Question 237

Original QCE Vault practice · 9 marks

Q237 · Practice questionTechnology-freeComplex unfamiliar9 marks

QUESTION 237 (9 marks)

Independent samples from a normal population with mean μ\mu and standard deviation 10 have sizes 25 and 100, with means X‾1,X‾2\overline X_1,\overline X_2. Consider U=(X‾1+X‾2)/2U=(\overline X_1+\overline X_2)/2 and pooled mean P=(25X‾1+100X‾2)/125P=(25\overline X_1+100\overline X_2)/125. For independent A,B, use Var⁡(aA+bB)=a2Var⁡(A)+b2Var⁡(B)\operatorname{Var}(aA+bB)=a^2\operatorname{Var}(A)+b^2\operatorname{Var}(B).
a)
Determine the mean and variance of U.
[3 marks]
b)
Determine the mean and variance of P, and identify the more precise estimator.
[3 marks]
c)
For 0≤a≤10\le a\le1, define the unbiased estimator Ea=aX‾1+(1−a)X‾2E_a=a\overline X_1+(1-a)\overline X_2. Use calculus to find the value of a giving least variance and relate the result to P.
[3 marks]
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