(1) μ=70, σ=7.
(1.1) Since the mean is 70 the areas under the normal curve above and below 70 are the same: 50%.
(1.2) If X=65, Z=(65-70)/7=-5/7=-0.7143; if X=70 Z=0. Probabilities are respectively 23.75% and 50%, so the probability of a score between 65 and 70 is 26.25% (area under the curve). This is equivalent to a Z-score of -0.6356, equivalent to a student score of 65.55.
(1.3) 26.25% approx.
(1.4) 50+26.25=76.25%.
(2) μ=67, σ=8.
(2.1) 50%
(2.2) Z=(75-67)/8=1 for a student score of 75, a probability of 84% (for less than 75) which is 84-50=0.34=34% above the mean. Z=1 means one standard deviation above the mean.
(2.3) 34%.
(2.4) Percentage for 75+ is 16%; 34+16=50%.