Question

Suppose that you are taking a course. There are two midterms and a final exam. Each midterm impacts 25% of the course grade while final exam impacts 50% of the grade. The first and second midterm scores follow a normal distribution with mean 84 points and the standard deviation of 9 points and mean 85 points and the standard deviation of 6. Assume that the final exam is also normally distributed with mean 87 and standard deviation of 6 points. Let W be the overall grade you get from this class.

1. What is the distribution of W? State the name and parameter values of the distribution.
2. What is the probability that a randomly chosen student gets A from this class? (You can assume55 that you have to get at least 90 to score an A).
3. What is the probability that a randomly chosen person’s grade is at within approximately 4points of the general class average? (You can also use the characteristics of standard normal here.)

Answer 1

here let x1,x2 and x3 are scores on three exams

therefore W =0.25x1+0.25x2+0.5x3

a)

distribution of W is normal with below parameter

mean =0.25*84+0.25*85+0.5*87=85.75

and std deviation =sqrt((0.25*9)²+(0.25*6)²+(0.5*6)²)=4.0389

b)

probability that a randomly chosen student gets A from this class =P(X>90)=P(Z>(90-85.75)/4.0389)=P(Z>1.05)

=0.1469

c) probability that a randomly chosen person’s grade is at within approximately 4points of the general class average

=P(-4/4.0389<Z<4/4.0389)=P(-0.99<Z<0.99)=0.8389-0.1611 =0.6778