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. $\displaystyle \mu = 0$

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

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

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
$\displaystyle \mu = 0$

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

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

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 $\displaystyle \sigma^2$?