SOLUTION: Please help me solve this problem : A set of examination marks are normally distributed with a mean of 75 and standard deviation of 5. If the top 5% of students got grade A and

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Question 894078: Please help me solve this problem :
A set of examination marks are normally distributed with a mean of 75 and standard deviation of 5. If the top 5% of students got grade A and the bottom 25% got grade F. Then what is the lowest mark for A and the highest mark for F
Thanks in advance.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
you need to find the z-score that has 5% of the area under the normal distribution curve to the right of it.

you also need to find the z-score that has 25% of the area under the normal distribution curve to the left of it.

you can use a z-score table or you can use a z-score calculator.

the best calculator i've been able to find is at the following link:

http://davidmlane.com/hyperstat/z_table.html

this calculator takes all the drudgery out of finding the z-score or finding the area to the left or right of a given area.

not only does it show you the z-score, but it also shows you a picture of the normal distribution curve and where under that your desired area lies.

using this calculator, i found that the z-score that had 5% of the area under the normal distribution curve to the right of it is 1.645.

here's a picture of what the calculator showed me.

i also found that the z-score that had 25% of the area under the normal distribution curve to the left of it is -.674

here's a picture of what the calculator showed me.

you now need to translate the z-score to a raw score.

the formula for z-score is:

z = (x-m)/s

z is the z score.
x is the raw score.
m is the population mean.
s, in this case, is the population standard deviation since you are not dealing with a sample but are dealing with the poopulation itself.

since you are solving for the raw score, you need to modify the formula to solve for x rather than z

start with z = (x-m) / s
multiply both sides of this equation by s to get:
z*s = x-m
add m to both sides of this equation to get:
z*s + m = x
slip sides to get:
x = z*s + m

that's the formula you need to find the raw score.

x = z*s + m

x is what you want to find.
z is equal to 1.645 or -.674.
s is equal to 5
m is equal to 75

let's find the raw score associated with a z score of 1.645

the formula becomes:

x = 1.645 * 5 + 75 which results in:

x = 83.225

let's find the raw scire assicuated with a z score of -.674

the formula becomes:

x = -.674 * 5 + 75 which results in:

x = 71.63

if you get a score below 71.63, you will be in the bottom 25% of the students who took the test.

if you get a score above 83.225, you will be in the top 5% of the students who took the test.

the z score calculator is so good, you could also have used the raw score rather than the z score.

i didn't show you how to do that because i wanted to deal with z scores which is what you have to deal with anyway.