The professor of a introductory calculus class has stated th

The professor of a introductory calculus class has stated that, historically, the distri- bution of final exam grades in the course resemble a Normal distribution with a mean final exam mark of = 60% and a standard deviation of = 12%. Use at least 3 decimals in your answers. (b) In order to pass this course, a student must have a final exam mark of at least 50%. What proportion of students will not pass the calculus final exam? (d) Suppose this professor randomly picked 10 final exams, observing the earned mark on each. What is the probability that 3 of these have a final exam grade of less than 50%?

Solution

Given mean final exam mark = 60% and standard deviation (SD)= 12%

b) to pass this course a student must have a final exam mark of at lest 50%

Let X be final exam mark then student need X>=50%

P(X>=50%) = P[(X-mean/SD)>=(50%-60%)/12%]= P(z>=-0.8333)

=1-P(Z<0.833) =1-0.202 = 0.798

d) from (b) probability thata student must have a final exam mark of at least 50% is 0.798

the student must have less than 50% is 1-0.798 = 0.202

Randomly picked 10 fianl exams consider as 10 trails each with probability of success ( successs means student get final exam grade of less than 50% ) is 0.202 and the number of success follows binomial distribution.

Probability that 3 of these have a final exam grade of less than 50% is

= (10C3) (0.202)^3 * (0.798)^7 = 0.204