Consider the performance analysis problem where you are interested in 
the service time (y) of database transactions. Let us assume that the
mean for this population (i.e. mean service time) is mu=100ms with
a standard deviation of sigma=20ms.  

Considering the sampling distribution of ybar, answer the following:

a) For samples of size n=400, what percentage of samples would you expect
   to result in sample means greater than 103ms. 

   Soln: Again, the sample size is large (i.e. n>=30) and so, as in a), 
         mu(ybar)=mu=100 and sigma(ybar)=sigma/sqrt(n)=20/sqrt(400)=1.
         We also know that the ybars are normally distributed. Hence
 	 z=(x-mu(ybar))/sigma(ybar)=(103-100)/1=3. From the empirical
         rule we know that the desired proportion is 50-(99.73/2)=0.135%.

b) In this case, let the sample size n=36. What percentage of samples 
   would you now expect to result in sample means greater than 103ms. 
   Make any appropriate assumptions.

   Soln: The sample size is large (i.e. n>=30) and so we know that
         mu(ybar)=mu=100 and sigma(ybar)=sigma/sqrt(n)=20/sqrt(36)=3.333.
         We also know that the ybars will be normally distributed 
         Hence z=(x-mu(ybar))/sigma(ybar)=(103-100)/3.333=0.9 and so 
         the desired proportion is 18.41%.