This problem has me scared to my wits, mostly because there are more like it!
The mean of a random sample of n observations drawn from a N $ ( \mu ,\sigma^{2} ) $ distribution is denoted by $ \overline{\chi} $. Given that P $ (| \overline{\chi} - \mu| > 0.5\sigma ) \lt 0.05 $
Find the smallest value of n.
So I tried:
Since $\overline{\chi}$ ~ N $ \left ( \mu ,\frac{\sigma^2}{n} \right ) $
P $ (| \overline{\chi} - \mu| > 0.5\sigma ) \lt 0.05 $ Should be:
$$ \frac{0.5\sigma - \mu}{\sqrt{\sigma^{2}/n}} \lt 0.05$$
$$1-\phi\left(\frac{0.5\sigma - \mu}{\sqrt{\sigma^{2}/n}}\right) \lt 0.05 $$
($\phi$ is the normal distribution function, I lookup a value from the normal dist. table.)
$$ \frac{0.5\sigma - \mu}{\sqrt{\sigma^{2}/n}} \lt 1.645$$
Now where do I go from here!?
This is totally confusing me, especially that darn modulus. How do I use my result so far with this?
P $ (| \overline{\chi} - \mu| > 0.5\sigma )$
Thanks Gideon