In case you're not already aware, the expression is called the "difference quotient" and represents the average rate of change of a function
over an interval
.
For the function , by substituting
we get
Then the difference quotient is
where the last equality holds as long as .
The mean is and standard deviation is
. Then the variable
.
Use substitution , then the variable
.
The probability can be calculated in the following way:
for X=80, and
. From the diagram
.
Conclusion: if 188 students take the exam, then 188·0.84=157.92 (157) would receive a score above 80