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 between 98ms and 102ms. 

   Soln: Since the sample size is large (i.e. n>=30) we know that
         mu(ybar)=mu=100 and sigma(ybar)=sigma/sqrt(n)=20/sqrt(400)=1.
         We also know that the ybars are normally distributed. Hence,
         using the transformations rule z=(x-mu(ybar))/sigma(ybar),
 	 z1=(98-100)/1=-2 and z2=(102-100)/1=2. From the empirical
         rule we know that the desired proportion is 95.45%.

b) 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%.

c) In this case, let the sample size n=26. 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 small (i.e. n<30) and so we know that
         mu(ybar)=mu=100 and sigma(ybar)=sigma/sqrt(n-1)=20/sqrt(25)=4.
         We also know that the ybars will be Student t distributed 
         with n-1=25 degrees of freedom if we assume that the service 
         times (i.e. y) are normally distributed. Hence, with this
 	 assumption, t=(x-mu(ybar))/sigma(ybar)=(103-100)/4=0.75. 
         From the t table, traversing the row for df=25, 0.75 is
         between 0.68 and 1.32, hence the desired proportion is
         between 10% and 25%.