The monthly proportion of purchases paid by check to a large grocery store that are returned because of insufficient funds can be viewed as the outcome of a Beta \((1,20)\) distribution.

(a) Simulate 12 monthly proportions of returned checks using the following 24 outcomes from a Uniform \((0,1)\) computer random number generator:

(.6829,.4283,.0505,.7314,.8538,.6762,.6895,.9955,.2201, \(.9144,3982, .9574\)

\(.0801, .6117, .3706, .2936, .2799, .3900, .7533, .0113, .5659\) , \(.9063, .5029, .6385\)

(a, b)\) if \(\left(X_{1}, X_{2}ight)\) are independent gamma-distributed random variables with parameters \(

(a, \beta)\) and \(

(b, \beta)\), respectively.)

(b) Calculate the sample mean and sample variance, and compare them to the appropriate population mean and variance.