Consider the portfolio of legacy COBOL programs in the portfolio 
of some large financial institution. Assume that a "student t"
distribution with 9 degrees of freedom may be used to describe
the effort (man hours) spent maintaining each program in the 
portfolio over the past year. Assume also that the mean effort is 
280 with a std dev of 50.

a) Find the proportion of programs that required more than 360
   hours of maintenance.

   We need to use the transformation rule. Since t=(360-280)/50=1.6
   then we are interested in the proportion to the right of 1.6 for
   the standard student t with df=9. For the row indexed by 9, 1.6
   is between 1.38 and 1.83 and so the desired proportion is between 
   5% and 10%.

b) Find the proportion of programs that required between 280 and 
   360 hours of maintenance.

   Again, we need to use the transformation rule. Note however that
   the "student t" is symmetric about the mean and so the desired
   proportion is merely 50% minus the proportion more than 360. From
   part a) this proportion is between 5% and 10% and so the desired 
   proportion is between 40% and 45%.

c) What if the degrees of freedom was 40 instead of 9. How would your 
   answer to part a) and part b) be different. Justify your answer.

   Since a "student t" with df>=30 is equivalent to the corresponding
   normal distribution, we may use the z table to determine the 
   desired proportion.

   For part a), the z value would also be 1.6 and so the desired 
   proportion is 50%-(89.04/2)=5.48%. This is as expected since for 
   larger df the distribution has thinner tails and so we would expect 
   a smaller proportion.
   
   For part b) we use the z table again and the desired proportion 
   in this case is 89.04/2=44.52%. Again, this is expected since for
   larger df we expect thinner tails hence more measurements clustered
   around the mean.