Scores on a final exam administered to all calculus classes

Scores on a final exam administered to all calculus classes at a large university are normally distributed with a mean of 71.6 and a standard deviation of 27.75. A) What percentage of students taking the test made a score of at most 70? B) What is the probability that a student taking the test had a score between 75 and 83? C)What percentage of students taking the test made a score that was more than two standard deviations(above and below) away from the mean? D) What score represents the 65th percentile?

Solution

This question can be easily solved using the z score approach.

Let X be any score from the examination

Therefore, z= (X-71.6)/27.75

a)

Z= (70-71.6)/27.75 = -0.057

Corresponding p value = 0.4772

Therefore, 47.72% students scored at most 70

b) Probability that score of students was between 75 and 83 = p(x<83) - p(x<75)

Z values are calculated the same way and are = 0.41 and 0.12

Coresponding p values = 0.6594 and 0.5488

Therefore, required probability = 0.6594-0.5488 = 0.1106

C) Given Z>2 or Z<-2

Corresponding p value = 0.02275

Required percentage = (2*0.02275)*100% = 4.55%

D) 65th percentile implies p=0.65

Corresponding z value = 0.38

Now, (X-71.6)/27.75 = 0.38

Therefore, X= 82.145

Therefore, the score is 82.145

Note: All p and z values have been obtained from the standard normal distribution