Hi,

I have a problem regarding CLT:

"The lengths of mass produced nails are normally distributed with a mean of $\displaystyle 3cm $ and variance of $\displaystyle 0.01 cm^2$. 16 nails are randomly chosen and laid end to end.

What is the probability that its length exceeds 48.5cm?"

This is my working out, I'm not sure what i am doing wrong:

Let $\displaystyle T_{16} = X_1 + X_2 + .... + X_{16}$, where $\displaystyle X_i$ are identically and independently distributed variables.

So, each of $\displaystyle X_i $~$\displaystyle N(3,0.01^2) $

Thus, $\displaystyle T_n $~ $\displaystyle N(48, 0.04^2)$

We need to find $\displaystyle P(T_{16} > 48.5)$. So

$\displaystyle P(T_{16} > 48.5) = 1 - P( T_{16} < 48.5) = 1 - P( Z < (48.5 - 48) / (0.04) ) = 1 - P( Z < 12.5)$

Then, I'm stuck here. Any help will be greatly appreciated thanks