# findings mean, SD and approximate distribution

• Jun 30th 2009, 06:03 AM
findings mean, SD and approximate distribution
Let X1,X2,....,X81 be a random sample from a distribution (not necessarily normal) with mean$\displaystyle \mu=59$ and SD $\displaystyle \sigma=8$. Let $\displaystyle \bar{X}=\frac{1}{81}\Sigma^{81}_{i=1}X_{i}$

a) What is the mean of $\displaystyle \bar{X}$
b) What is the standard deviation of $\displaystyle \bar{X}$
c) What is the approximate distribution of $\displaystyle \bar{X}$

I'm not really sure what this question is asking. I know how to find means and standard deviations, but not from what is given in this question. Thank you.
• Jun 30th 2009, 07:15 AM
Random Variable
a) $\displaystyle \mu = 59$

b) $\displaystyle \frac {\sigma}{\sqrt{n}} = \frac{8}{9}$

c) $\displaystyle N\Big(\mu,\frac {\sigma^{2}}{n}\Big) = N\Big(59,\frac{64}{81}\Big)$
• Jun 30th 2009, 08:31 AM
The mean is the same but the SD is divided by $\displaystyle \sqrt{n}$? I never would have known that. What is that SD equation from? Also, how can you figure it is a normal distribution? It is probably obvious but I can't find info like that at all in my book and would like to learn.
• Jun 30th 2009, 09:28 AM
Random Variable
If $\displaystyle X_{1}, X_{2}, ... , X_{n}$ are indepedent random variables with respective means $\displaystyle \mu_{1}, \mu_{2}. ... , \mu_{n}$ and variances $\displaystyle \sigma^{2}_{1}, \sigma^{2}_{2}, ... , \sigma^{2}_{n}$

then the mean and variance of $\displaystyle Y = \sum^{n}_{i=1} a_{i} X_{i}$ are $\displaystyle \mu_{Y} = \sum^{n}_{i=1} a_{i} \mu_{i}$ and $\displaystyle \sigma^{2}_{Y} = \sum^{n}_{i=1} a_{i}^{2} \sigma^{2}_{i}$ respectively

You can prove the preceding directly from the definition.

Therefore, $\displaystyle \mu_{\bar{X}} = \sum^{n}_{i=1} \frac {1}{n} \mu = \frac {1}{n} (n \mu) = \mu$

and $\displaystyle \sigma^{2}_{\bar{x}} = \sum^{n}_{i=1} (\frac {1}{n})^{2} \sigma^{2} = \frac{1}{n^{2}} (n \sigma^{2}) = \frac {\sigma^{2}}{n}$

part(c) is the result of the central limit theorem