SOLUTION: Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.54 and a standard deviation of 0.42. Using the empir

Question 1190245: Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.54 and a standard deviation of 0.42. Using the empirical rule, what percentage of the students have grade point averages that are greater than 3.8? Please do not round your answer.
Found 2 solutions by Boreal, math_tutor2020:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
(Odd not to round the answer at the end when the empirical rule itself is rounded.). Anyway, 3.8 is
(3.8-2.54/4.2 or 3 sd s above the mean. This iii 0.013 % of all students.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Convert the raw score x = 3.8 to its corresponding z score
z = (x-mu)/sigma
z = (3.8 - 2.54)/(0.42)
z = (1.26)/(0.42)
z = 3
The raw score is 3 standard deviations above the mean.

Empirical Rule:

All values mentioned in that image are approximate.

According to the Empirical Rule, roughly 99.7% of the distribution is within 3 standard deviations of the mean.
This leaves 100% - 99.7% = 0.3% for both tails
which cuts in half to (0.3%)/2 = 0.15% for each tail
Roughly 0.15% of the entire area under the standard normal Z curve is to the right of z = 3.
There's roughly a 0.15% chance of picking someone with a grade point average higher than 3.8

Answer: 0.15%

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