Question #1:

For some population, the measurement of interest is known to be 
normally distributed with mean mu=20 and standard deviation
sigma=5.

Determine the proportion of measurements in the population that
satisfy the following: 

a) Between 15 and 25

   From the empirical rule we know that 68.27% of measurements will
   be within one standard deviation of the mean. This means that
   68.27% will be between 15 and 25.
 
b) Greater than 25

   From the empirical rule we know that 68.27% of measurements will
   be within one standard deviation of the mean. This means that
   68.27% will be between 15 and 25. By symmetery, since 20 is the 
   mean, this also means that (68.27/2)=34.135% will be between 20 
   and 25. But we are interested in the proportion greater than 25.
   Since symmetry also tells us that 50% will be greater than 20 then
   the desired proportion is 50% - 34.135% which is 15.865%.

c) Less than 10

   From the empirical rule we know that 95.45% of measurements will
   be within two standard deviations of the mean. This means that
   95.45% will be between 10 and 30. By symmetery, since 20 is the 
   mean, this also means that (95.45/2)=47.725% will be between 10 
   and 20. But we are interested in the proportion less than 10.
   Since symmetry also tells us that 50% will be less than 20 then
   the desired proportion is 50% - 47.725% which is 2.275%.

d) Between 5 and 25

   From the empirical rule we know that 68.27% of measurements will
   be within one standard deviation of the mean. This means that
   68.27% will be between 15 and 25. By symmetery, since 20 is the 
   mean, this also means that (68.27/2)=34.135% will be between 20 
   and 25. 
   From the empirical rule we know that 99.73% of measurements will
   be within three standard deviations of the mean. This means that
   99.73% will be between 5 and 35. By symmetery, since 20 is the 
   mean, this also means that (99.73/2)=49.865% will be between 5 
   and 20. 
   Hence, 34.135%+49.865%=84% is the desired proportion.

e) Between 20 and 30

   From the empirical rule we know that 95.45% of measurements will
   be within two standard deviations of the mean. This means that
   95.45% will be between 10 and 30. By symmetery, since 20 is the 
   mean, this also means that (95.45/2)=47.725% will be between 20 
   and 30. 

Question #2:

Consider the population of C++ programs in some portifolio. Let 
the measurement of interest be program size (i.e. the number of
lines of code in each program). Assume that program size is normally 
distributed with mean mu=400 and standard deviation sigma=50.

Determine the proportion of programs in the portfolio that are:

a) Less than 300 lines of code.

   From the empirical rule we know that 95.45% of measurements will
   be within two standard deviations of the mean. This means that
   95.45% of programs will be between 300 and 500 lines of code. 
   This also means that (95.45/2)=47.725% will be between 300 and 
   400 lines of code. But we are interested in the proportion less 
   than 300 lines of code. Since symmetry also tells us that 50% will 
   be less than 400 then the desired proportion is 50% - 47.725% which 
   is 2.275%.

b) Greater than 400 lines of code.

   By symmetry 50% of the programs will be greater than the mean, 
   that is 400 lines of code.

c) Between 450 and 550 lines of code.

   From the empirical rule we know that 68.27% of measurements will
   be within one standard deviation of the mean. This means that
   68.27% will be between 350 and 450. By symmetery, since 400 is the 
   mean, this also means that (68.27/2)=34.135% will be between 400 
   and 450. 
   From the empirical rule we know that 99.73% of measurements will
   be within three standard deviations of the mean. This means that
   99.73% will be between 250 and 550. By symmetery, since 400 is the 
   mean, this also means that (99.73/2)=49.865% will be between 400 
   and 550. 
   Hence, 49.865%-34.135%=15.73% is the desired proportion.