1. A large statistics class took an exam. The scores followed a normal curve with mean 78 and
standard deviation 10. Students who scored 90 or above got an A on the exam. Those who scored
between 80 and 90 got a B, between 70 and 80 a C, between 60 and 70 a D, and below 60 an F.
a. Estimate the percentage of students who earned an A on the exam.
b. Estimate the percentage of students who earned a B on the exam.
c. Estimate the percentage of students who earned a C on the exam.
d. Estimate the percentage of students who earned a D on the exam.
e. Estimate the percentage of students who earned an F on the exam.
Please show as much work as you can and don't round. Thank you.
Solution
http://www.math.umt.edu/elias/NormalDistributionTable.jpg
given is the normal distribution table
= 78
= 10
1) for students scoring A : we find the area under the curve of standard normal distribution using
Z value
Z for x=90 : (90-78)/10 = 1.2
(z)=(1.2)=.8849
so the required area is 1-(z)= 1=0.8849= .1151
hence the percentage will be 11.51%
2) for students scoring B : we find the area under the curve of standard normal distribution using
Z value
Z for x=80 : (80-78)/10 = 0.2
(z)=(0.2)=.5793
% students scoring above B = (1-.5793)*100=42.07%
% students scoring B = % students scoring above B -% students scoring A
= 42.07-11.51

2. = 30.56%
3) for students scoring C : we find the area under the curve of standard normal distribution using
Z value
Z for x=70 : (70-78)/10 = -0.8
(z)=(-0.8)=.2119
% students scoring above C = (1-.2119)*100=78.81%
% students scoring C = % students scoring above C -% students scoring above B
= 78.81-42.07
= 36.74%
4) for students scoring D : we find the area under the curve of standard normal distribution using
Z value
Z for x=60 : (60-78)/10 = -1.8
(z)=(-1.8)=.0359
% students scoring above D = (1-.0359)*100=96.41%
% students scoring D = % students scoring above D -% students scoring above C
= 96.41-78.81
= 17.60%
5) for students scoring F : we find the area under the curve of standard normal distribution using
Z value
Z for x=60 : (60-78)/10 = -1.8
(z)=(-1.8)=.0359
% students scoring F =100- % students scoring above D
% students scoring D = % students scoring above D -% students scoring above C = = 100 -
96.41
= 3.59%
Verification: Sum of all 5 percentages = 100