The distribution of grades in an introductory finance class is normally distributed, with an expected grade of 65. if the standard deviation of grades is 15, in what range would you expect 90.00 percent of the grades to fall

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.

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