Student Grades

Date: 08/13/97 at 23:27:21
From: Amy Hunter
Subject: Statistics

A professor at a local college noted that the grades of her students 
were normally distributed with a mean of 74 and a standard deviation 
of 10. The professor has informed us that 6.3 percent of her students 
received an A, while only 2.5 percent of her students failed the 
course and received F's.

What is the minimum score needed to make an A?  What was the maximum 
score among those who received an F?  And if there were 5 students who 
did not pass the course, how many students took the test?  

I can figure out the mean and the standard deviation, but am confused 
about what they want me to do, the kind of formula I should be using, 
and what lets me know that? Thanks.

Amy Hunter

Date: 08/14/97 at 07:46:04
From: Doctor Anthony
Subject: Re: Statistics

                       x - m
Use the formula   z = -------      m = mean = 74,    s = s.d. = 10
                         s

The Normal tables give total probabilities up to a value of z.  
Sometimes you need the area beyond a value of z (upper tail area), so 
look up area corresponding to the value of z you have calculated and 
subtract this from 1. If you are given the probability, then you read 
the table backwards, i.e. you enter the area and find the 
corresponding value of z.

(a) What is the minimum mark for an A?  Here we shall be reading the 
    table backwards. The upper tail area is 6.3 percent = .063, so 
    look up in table an area equal to 1 - .063 = 0.937  and you will 
    find this corresponds to  z = 1.53

                         x - 74
    So we have   1.53 = --------   and so   x = 74 + 10 x 1.53
                           10                 = 89.3

    So a mark of 90 percent would be the minimum for a grade A.

(b) What is the maximum mark of someone who failed?  Again we read the 
    table backwards, knowing that the tail area must be 2.5 percent = 
    0.025    Although this is the lower tail area, look up in the 
    table an area 1 - .025 = 0.975  You will find this corresponds to 
    a z value 1.96,  but we are dealing with values below the mean 
    (z = 0) and so z is negative, i.e. z = -1.96

                                   x - 74 
    Therefore we put   -1.96  =  ---------    so x = 74 - 10 x 1.96
                                     10            = 54.4

    So a mark of 54 percent would be the maximum of someone who 
    failed.

(c) If 5 students failed, how many took the test?

    Since 5 students corresponds to 2.5 percent of the total, simple 
    proportion tells us that the total taking the test is 

                5 x 100
                -------     = 200 students.
                  2.5

-Doctor Anthony,  The Math Forum
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