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Monash Bus Stats question

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lynt.br:
I'm having trouble understanding question (1)(b) from the sample exam we were given in the last revision lecture.

The question asks: For a simple random sample, X-bar is an unbiased estimator of μ if E[X-bar] = μ. Prove that this is true when X ~ N(μ,σ2). What does this mean in practice?

I'm don't quite follow what the question is asking so I'm not sure what exactly it is I'm trying to prove, or how I would go about doing so.

Thanks!

TrueTears:
The proof is a fairly easy mathematical exercise:

for some i.





since we are given ~ so

Thus as required.

The meaning of this is that whenever we take a random sample and calculate the sample mean we can be sure that the value we get is an unbiased estimate of the true population mean.

lynt.br:
Thanks for the reply but I still can't get my head around it -___-

Would you be able to explain those steps? That might help me a bit (I'm a bit of a lost cause when it comes to maths haha).

TrueTears:
haha alright, well the first line is just replacing with it's summation notation.

Second line... you can take out a factor of since the expectation function (E( )) is independent of constants.

Then we expand the summation of on the third line, then on fourth line we replace each with since it says in the question that X has an expected value, , of

But we have 'amount' of , so we have in total.

Then the cancels out the which leaves us with !

lynt.br:

--- Quote from: TrueTears on November 06, 2010, 03:41:45 am ---haha alright, well the first line is just replacing with it's summation notation.

Second line... you can take out a factor of since the expectation function (E( )) is independent of constants.

Then we expand the summation of on the third line, then on fourth line we replace each with since it says in the question that X has an expected value, , of

But we have 'amount' of , so we have in total.

Then the cancels out the which leaves us with !

--- End quote ---

Ahhh it's like a light just flicked on in my head haha! Thanks for having patience with me, this made it a lot clearer!

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