Suppose that, for a certain mathematics class, the scores are normally distributed with a mean of 75 and a standard deviation of 9. The teacher wishes to give A's to the top 7% of the students and F's to the bottom 7%. The next 17% in either direction will be given B's and D's, with the other students receiving C's.
What is the cumulative distribution?
Grade Percent Cumulative
A 7%
%
B 17%
%
C
%
%
D 17%
%
F 7%
%
How many students were there? n = ?
C = mean ± 26%
To determine the cumulative distribution for each grade, we need to find the cutoff points based on the normal distribution.
First, let's find the z-scores corresponding to the percentiles for each grade category:
For A grade (top 7%):
The area to the left of the cutoff z-score is 1 - 7% = 93%
Using a standard normal distribution table or a calculator, we can find the z-score corresponding to the cumulative area of 0.93, which is approximately 1.48.
For B grade (next 17% in either direction):
The area to the left of the left cutoff z-score is 7% + 17% = 24%
The area to the right of the right cutoff z-score is 7% + 17% = 24%
So, each tail contains 12% (half of 24%).
The cumulative area to the left of the left cutoff z-score is 7% + 12% = 19%
The cumulative area to the right of the right cutoff z-score is 93% - 12% = 81%
Using a standard normal distribution table or a calculator, we can find the z-scores corresponding to these cumulative areas, which are approximately -0.88 and 0.88.
For C grade:
The area to the left of the left cutoff z-score is 7% + 17% + 17% = 41%
The cumulative area to the left of the right cutoff z-score is 93% - 17% - 17% = 59%
Using a standard normal distribution table or a calculator, we can find the z-scores corresponding to these cumulative areas, which are approximately -0.25 and 0.25.
For D grade (next 17% in either direction):
The area to the left of the left cutoff z-score is 7% + 17% + 17% + 17% = 58%
The area to the right of the right cutoff z-score is 7% + 17% + 17% + 17% = 58%
So, each tail contains 29% (half of 58%).
The cumulative area to the left of the left cutoff z-score is 41% + 29% = 70%
The cumulative area to the right of the right cutoff z-score is 93% - 29% = 64%
Using a standard normal distribution table or a calculator, we can find the z-scores corresponding to these cumulative areas, which are approximately -0.68 and 0.68.
For F grade (bottom 7%):
The area to the right of the cutoff z-score is 7%
Using a standard normal distribution table or a calculator, we can find the z-score corresponding to the cumulative area of 0.07, which is approximately -1.48.
Now we can calculate the cumulative distribution for each grade:
Grade Percent Cumulative
A 7% 93%
B 17% 24% (left cutoff) to 81% (right cutoff)
C 41% (left cutoff) to 59% (right cutoff)
D 17% 70% (left cutoff) to 64% (right cutoff)
F 7% 7%