Question #1:

Find the area under the standard "student t" curve:

Note that this could be rephrased "Find the proportion of observations
that satisfy the following if the distribution is standard student t".

a) for df=18, to the right of 1.25

   From the standard "student t" table we traverse the row indexed by
   df=18 and notice that 1.25 is between 0.688 and 1.33 hence we observe
   from the corresponding column headings that the desired proportion 
   is between 10% and 25%.

b) for df=10, to the left of -0.7

   Since the "student t" distribution is symmetric about the mean this
   problem is equivalent to determining the proportion to the right of
   0.7. From the standard "student t" table we traverse the row indexed 
   by df=10 and notice that 0.7 is in the table and so the desired 
   proportion is the corresponding column heading, that is 25%.

c) for df=15, to the left of 0.8

   From the standard "student t" table we traverse the row indexed by
   df=15 and notice that 0.8 is between 0.691 and 1.341 and, observe
   from the column headings that the corresponding proportions are
   25% and 10%. Since we need the proportion to the left, the desired
   proportion is between 75% and 90%. 



Question #2:

For some population, with mean=150 and std dev=10, assume that the 
"student t" distribution with 20 degrees of freedom is appropriate.

a) Find the proportion of measurements less than 150.

   This may be directly answered by recalling that the distribution is
   symmetric about the mean and so the answer in this case is clearly
   50%

b) Find the proportion of measurements less than 140.

   We need to use the transformation rule. Since t=(140-150)/10=-1
   then we are interested in the proportion to the left of -1 for
   the standard student t with df=20. By symmetry, this is equivalent
   to the proportion to the right of 1 and so from the row indexed
   by 20 we see that the desired proportion is between 10% and 25%.



Question #3:

For some population, with mean=50 and std dev=10, assume that the 
"student t" distribution with 173 degrees of freedom is appropriate.
Find the proprortion of measurements less than 60.

    Since we consider the "student t" to be equivalent to the 
    normal for df>=30, we may consider this population to be 
    normally distributed and proceed. Two approaches to solving
    this problem are presented below:

    Given this observation, notice that 60 is one std dev from
    the mean and so, since normality is appropriate, then the
    empirical rule applies and so the desired proportion is
    50% + 68.27/2 = 84.135%.

    Alternatively, from the standard normal table, the desired
    proportion is 84.13%.