1) The Z score for a value of 80 when the mean is 75 and the standard deviation is 4.2 is:
z = (80-75)/4.2 = 1.19
Look up the cumulative distribution value for 1.19 (here is an online calculator
, leave the mean as zero and the standard deviation as one). You get 88.3%.
2) Find the z score for 98% (2% failing). You can use the same calculator from above by clearing out the z score, and putting .98 in for the cumulative probability. You get 2.054. Using the formula above:
2.054 = (80 - x)/4.2
x = 73.5
3) In order to get 2%, you need a z score of 2.054, so using the same formula
2.054 = (80-75)/x where x is the new standard deviation.
x = 2.43