- #1

FredericChopin

- 101

- 0

## Homework Statement

"Consider a standard uniform density. The mean for this density is .5 and the variance is 1 / 12. You sample 1,000 sample means where each sample mean is comprised of 100 observations. You take the standard deviation of the 1,000 sample means. About what number would you expect it to be?

A: [itex] \frac{1}{12} [/itex]

B: [itex]\frac{1}{ \sqrt{12*100} } [/itex]

C: [itex] \frac{1}{(12*100)} [/itex]

D: [itex]\frac{1}{ \sqrt{12*1000} } [/itex]"

## Homework Equations

[tex]Var( \bar{X} ) = \frac{ \sigma ^{2} }{n} [/tex]

## The Attempt at a Solution

If the variance of a sample mean with

*n*observations is [itex]Var( \bar{X} ) = \frac{ \sigma ^{2} }{n} [/itex], then that would mean that the standard deviation of a sample mean with

*n*observations is [itex] \sqrt{ \frac{ \sigma ^{2} }{n} } [/itex], which is [itex] \frac{ \sigma }{ \sqrt{n} } [/itex].

But in this problem, there are 1000 sample means each with 100 observations. In general, this is a question about finding the standard deviation of

*m*sample means of

*n*observations each.

I'm puzzled that there is no option for the answer that since there are a total of 1000*100 observations, the standard deviation is [itex] \sqrt{ \frac{ \frac{1}{12} }{1000*100} } [/itex]. So I tried the fact that there are 1000 sample means, so I answered [itex] \sqrt{ \frac{ \frac{1}{12} }{1000} } [/itex], but it got marked incorrect. Is it possible because there are 100 observations for each sample mean, regardless of how many there are, that the answer is [itex] \sqrt{ \frac{ \frac{1}{12} }{100} } [/itex]? And in that case, why are the previous two answers I gave incorrect?

Thank you.