Question

Suppose that the score in a qualifying exam for a certain profession is normally distributed with mean 80 and standard deviation 36. Use Excel to find the following probabilities:

b) Find the probability that a student gets a score between 50 and 100.

c) To get an A, a student must score in the top 8%. What is the minimum score needed to get an A?

d) Find the probability that the average score in a sample of 60 students is between 50 and 100?

e) Graph the distributions of the random variable in c) and d) and explain why the answers in b) and d) are different.

Please answer using excel functions.

Answer 1

Solution:

    We are given that: the score in a qualifying exam for a certain profession is normally distributed with mean 80 and standard deviation 36.

We have to use Excel functions to find following probabilities:

Part b) Find the probability that a student gets a score between 50 and 100.

That is: P( 50 < X < 100 ) = ........?

Following is the Excel function:

=NORM.DIST( x , mean , SD, Cumulative)

Since we have to find probability between two values, we do subtraction as:

=NORM.DIST( 100 , 80 , 36 , TRUE) - NORM.DIST( 50 , 80 , 36 , TRUE)

= 0.508414

= 0.5084

Thus : P( 50 < X < 100 ) = 0.5084

Part c) To get an A, a student must score in the top 8%. What is the minimum score needed to get an A?

That is find x value,such that:

P( X > x) = 8%

P( X > x) = 0.08

Since this 8% area is in right tail, we need to do 1 - 0.08 in excel command for probability.

=NORM.INV( 1 - 0.08 , 80 , 36 )

=130.58

=131

That is:

P( X > 131 ) = 8%

Part d) Find the probability that the average score in a sample of 60 students is between 50 and 100?

Sample Size = n = 60

Thus we have to find:

For finding probability of sample mean, we use Standard Error instead of Standard deviation.

Thus Standard Error is given by:

Thus we use following excel command:

=NORM.DIST( 100 , 80 , 4.6476 , TRUE) - NORM.DIST( 50 , 80 , 4.6476 , TRUE)

= 1.0000

That is:

Part e) Graph the distributions of the random variable in c) and d) and explain why the answers in b) and d) are different.

graph of part c)

graph of part d)

        There is difference between part b and part d, since in part b) we found probability of individual x value which uses standard deviation = 36 whereas in part d) we found probability for sample mean and we use standard error which is 4.6476

So due to increase in sample size, there is decrease in variation and hence probability of sample mean covering area between 50 and 100 is more than that of individual x value.