Question 1170017: A teacher gave a test on which the students' marks were normally distributed, but the results were pathetic. The mean was 52%, and the standard deviation was 12%. The teacher decided that the top 10% of the students should get A's, the next 20% should get B's, the next 40% should get C's, the next 20% should get D's, and the bottom 10% should get F's. To the nearest percent, what was the minimum percent a student had to achieve in order to recieve a B? Do not include the percent sign in your answer.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! mean = 52%
standard deviation = 12%
top 10% get A
next 20% get B
next 40% get C
next 20% get D
bottom 10% get F
minimum B score will be the z-score that has 10% + 20% = 30% of the area under the normal distribution curve to the right of it.
the table are set to calculate off of ratios, not percents.
just divide the percent by 100 and you get an area of .30 to the right of the z-score.
since the tables only look for area to the left of the z-score, you are looking for a z-score that has 1 - .30 = .70 of the area under the normal distribution curve to the left of it.
if you look in the z-score table, you will find that a z-score of .52 has .69847 of the area under the normal distribution curve to the left of it and a z-score of .53 has .70194 of the area under the normal distribution curve to the left of it.
a manual interpolation leads to a z-score of .524409 rounded to 5 decimal digits.
that's pretty close to having an area of .7 to the left of it.
a check with an online calculator shows that a z-score of .524409 has an area of .7 to the left of it.
the display of the calculator results are shown below.
the calculator i used can be found at https://www.calculator.net/z-score-calculator.html?c1raw=3.075&c1mean=3.2&c1sd=.1&calctype=zscore&x=81&y=18
to find the raw score associated with that, use the z-score formula of:
z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard deviation
the formula becomes .524409 = (x - 52) / 12.
solve for x to get x = .524409 * 12 + 52 = 58.292908.
that's the lowest score you can get and still get a B.
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