Those grades that do NOT fall in that range would be 10% of the grades; half of 10%, or 5%, would be below a certain grade, and the other 5% would be above a different certain grade. Again, we have to determine the range, i. e., the lowest grade and the highest grade.
I have it easy because my TI-83 Plus calculator will give me the z-score for the lowest 5% of the spread of grades. That z-score is -1.645. z = +1.645 corresponds to the upper limit we were taking about.
Our job now is to take these z-scores, as well as the mean and std. dev., and determine the actual grades that limit the middle 90% of the grades.
x - 65
First, if z = -1.645 (leftmost 5% of grades), then -1.645 = -----------
15
Here, x is the lower end point of the middle 90% of grades.
Solving this equation for x, (15)(-1.645) = x - 65. Solving this for x,
x = 40. The lower end point of the middle 90% of grades is 40. That is 25 less than the mean (65). The upper end point would be 65+25 = 90.
We would expect 90% of the grades to fall between 40 and 90.
