Question #1:

Find the area under the standard normal curve:

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

a) to the right of 1.25

   From the standard normal table we know that 78.87% of observations are
   between -1.25 and 1.25. Since the normal curve is symmetric, and we are
   only interested in the proportion of observations greater than 1.25, the
   desired proportion is 50 - (78.87)/2 = 10.565%

b) to the left of -0.4

   The argument is as above. In this case 31.08% of observations are
   between -0.4 and 0.4. Since we are interested in the proportion to the
   left of -0.4, the desired proportion is 50 - (31.08)/2 = 34.46%

c) to the left of 0.8

   This could be approached in several ways. The usual way we would solve
   this problem is to recognize that the proportion to the left of 0.8 is
   the sum of the proportion less than mu and the proportion between mu
   and 0.8. From the standard normal table, the proportion between -0.8
   and 0.8 is 57.63% so by symmetry the proportion between mu and 0.8 is
   57.63/2 hence the desired proportion is 50 + (57.63)/2 = 78.815%.

   Alternatively: (Note: skip unless you are curious to see another approach.)  
   Another way of solving this is to think of the proportion to the left of
   0.8 as the sum of the proportion between -0.8 and 0.8 and the proportion
   to the left of -0.8. The proportion between -0.8 and 0.8 is 57.63% and
   the proportion to the left of -0.8 is 50 - (57.63)/2 = 21.185%. Hence,
   using this approach, the desired proportion is also 78.815%.

d) between 0.4 and 1.3

   This can also be worked several ways but this time I will present one
   solution. We can first compute the proportion between -1.3 and 1.3
   and then subtract the proportion between -0.4 and 0.4. This will give
   us the proportion between 0.4 and 1.3 plus the proportion between
   -1.3 and -0.4. We are only interested in the proportion between
   0.4 and 1.3 but, because of symmetry, we can simply divide by 2. So, 
   since the proportion between -1.3 and 1.3 is 80.64% and that between
   -0.4 and 0.4 is 31.08%, our desired proportion is (80.64 - 31.08)/2 = 
   24.78%

   Exercise: Solve using the approach discussed in class to see that you
   obtain the same answer (i.e. notice that you may subtract the proportion
   between mu and 0.4 from the proportion between mu and 1.3).

e) between -0.3 and 0.9

   Again several possible approaches. I will outline one and leave it 
   up to you to complete the computation and present another. First,
   you could compute the proportion to the left of 0.9 and subtract the
   proportion to the left of -0.3. Alternatively, you could compute
   the proportion between 0 and -0.3 and add it to the proportion between
   0 and 0.9. The proportion between 0 and -0.3 is simply the proportion
   between -0.3 and 0.3 divided by 2. That is 23.58/2 = 11.79%. The
   proportion between 0 and 0.9 is the proportion between -0.9 and
   0.9 divided by 2. That is 63.19/2 = 31.595%. Hence the desired 
   proportion is 43.385%. 

f) outside -1.5 to 1.5

   This is simply 100 minus the proportion between -1.5 and 1.5. That is
   100 - 86.64 = 13.36%.