Question:

A colleague majoring in IS is interested in using regression methods
to develop an equation to predict number of change requests from module
size. She designs an experiment, collects data and computes the
following statistics:

       size (lines of code): mean = 1600; Std Dev = 200
       # of change requests: mean = 30; Std Dev = 5
       correlation between size and # of change requests = 0.8

Assume normality where necessary.

1. What proportion of programs had less than 30 change requests.
 
2. For programs of size 1600, what proportion had less than 30 change 
   requests.

3. Estimate the number of change requests for programs of size 1900.
 
4. For programs of size 1900, what proportion had less than 30 change 
   requests.
 

Solution:

1. Since we are not interested in programs of a particular size,
   regression is not required. The required mean in this case is
   the mean of all programs (i.e. 30) and so the proportion is
   50%.

2. In this case we are interested in programs of a particular size
   and so regression is applicable. Note however that the size
   is 1600 and since this is also the overall mean for size then
   by the regression property that says that (ybar, xbar) is on
   the regression line we know that the required mean is also 30
   and so the proportion is again 50%.

3. In this case we are interested in programs of a particular size
   and so need to determine the mean number of change requests
   for this size. Notice also that since we are interested in a
   particular size then size will be x.
                   rise=run[r(Sy/Sx)]
                       =300[0.8(5/200)]
                       =6
   Hence the required mean, and so the desired estimate, is 30+6=36.

3. In this case we are interested in programs of a particular size
   but since we know the mean for this size from Q3 we only need to
   compute the std dev before applying normal theory.
        STEP 1: Determine mean.
                   mean=36 change requests
        STEP 2: Determine std dev.
                   Sy|x=Sy(sqrt(1-r^2))
                       =5(sqrt(1-0.8^2))
                       =3
        STEP 3: Assume normality and determine z.
                   z=(30-36)/3=-2
   Hence desired proportion is (100-95.45)/2=2.275%