1. ## Hypothesis testing question

In a large city the distribution of incomes per family has a standard deviation of £2500.
For a random sample of 400 families, what is the probability that the sample mean per family is within £500 of the actual income per family?

2. $
\mu = 0
$

$
E\left(\frac{1}{n}\sum x_i^2\right) = \sigma^2
$

$
Var\left(\frac{1}{n}\sum x_i^2\right) = E\left[\left(\frac{1}{n}\sum x_i^2 - \sigma^2\right)^2\right] = ... = \frac{2\sigma^4}{n}
$

This should get you started, I hope...

I'm on my way to work, so I don't have much time, but let me know if you need help with (...)

3. Originally Posted by temp
$
\mu = 0
$

$
E\left(\frac{1}{n}\sum x_i^2\right) = \sigma^2
$

$
Var\left(\frac{1}{n}\sum x_i^2\right) = E\left[\left(\frac{1}{n}\sum x_i^2 - \sigma^2\right)^2\right] = ... = \frac{2\sigma^4}{n}
$

This should get you started, I hope...

I'm on my way to work, so I don't have much time, but let me know if you need help with (...)

1) How do you know that E(the function) is $\sigma^2$?
2) How did you calculate the variance?

thank you!

4. see Estimation with applications to tracking and navigation By Yaakov Bar-Shalom, Xiao-Rong Li, Thiagalingam Kirubarajan p.106

Reference courtesy of physicsforum.com