Question

A random sample of size 80 is taken from a population that follows the following density function: x e 1280 Using the Central Limit Theorem, find the approximate probability that the sample mean exceeds 13.

Answer 1

for above is a gamma distribution for which parameter $\alpha$=3 and $\beta$=4

therefore mean =$\alpha$$\beta$ =12

ad std deviation =sqrt($\alpha$$\beta$²) =sqrt(48)=6.928

for sample size n=80 ;

estimated mean =12

and std error of mean =6.928/sqrt(80)=0.775

hence from central limit theorum ; approximate probability =P(Xbar>13)=1-P(Xbar<13)

=1-P(Z<(13-12)/0.7746)=1-P(Z<1.29)=1-0.9015=0.0985