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?
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?
$\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 (...)