Question 1146567: An English professor assigns letter grades on a test according to the following scheme.
A: Top 13% of scores
B: Scores below the top 13% and above the bottom 61%
C: Scores below the top 39% and above the bottom 22%
D: Scores below the top 78% and above the bottom 6%
F: Bottom 6% of scores
Scores on the test are normally distributed with a mean of 75.1 and a standard deviation of 9.1. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.
Answer by VFBundy(438)   (Show Source):
You can put this solution on YOUR website! "Scores below the top 78% and above the bottom 6%" is another way of saying scores between 6% and 22%.
The rest of the info given isn't needed.
First, we will find what the numerical score is for 22%.
Go to a z-table and find where the value is 0.22. (Or, the closest value that's below 0.22, since the question states these are scores BELOW 22%...or, as they put it, scores below the top 78%.) The z-score where this occurs is -0.78. (**NOTE: The fact that the z-score is -0.78 and the question states 78% is purely coincidental.)
Plug this into the following formula:
(x - mean)/SD = -0.78
(x - 75.1)/9.1 = -0.78
Solve for x:
(x - 75.1)/9.1 = -0.78
(-0.78)(9.1) = (x - 75.1)
-7.098 = x - 75.1
x = 68.002
So, the 22% score is 68 (rounded off).
Next, we will find what the numerical score is for 6%.
Go to a z-table and find where the value is 0.06. (Or, the closest value that's above 0.06, since the question states these are scores ABOVE 6%.) The z-score where this occurs is -1.55.
Plug this into the following formula:
(x - mean)/SD = -1.55
(x - 75.1)/9.1 = -1.55
Solve for x:
(x - 75.1)/9.1 = -1.55
(-1.55)(9.1) = (x - 75.1)
-14.105 = x - 75.1
x = 60.995
So, the 6% score is 61 (rounded off).
The numerical limits for a D grade are "higher than 61 and less than 68."
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