
1 Attachment(s)
Central Limit Theorem
See Question 3 on the attached document....
It is a past exam paper but the solutions are not provided so i cant see where i am going wrong....
I derived the PGF in part (i) then proceeded to get E(x) as 11/5 and Var(x) = 39/25
The problem is in part (iii) in the central limit theorem. I constructed the sample mean as (11/5)*50 to give 110 and the variance as $\displaystyle {\frac {\sigma}{\sqrt {n}}}$ which i got as such a small value (sqrt39/25)/(sqrt50) that it made my Z values so large that phi of these values were like 56 and 320 which would obviously be approximately 10, meaning almost the whole of the set is in this interval.... anyway it didnt seem right! have i gone wrong somewhere?

Have i gone wrong on part (iii)?

I'm not sure what you're asking.
I get $\displaystyle E(X)=2.2$ and $\displaystyle E(X^2)=6.4$
So $\displaystyle V(X)=1.56$
Thus $\displaystyle Y\approx N((50)(2.2), (50)(1.56))$
Since the X's are integer based you may want to put in a correction factor, but that won't
alter your answer a whole lot.