The grade point average GPA of students in the Engineering M

The grade point average (GPA) of students in the Engineering Management program at New York University is normally distributed with a mean of 3.20 and a standard deviation of 0.4.

What percentage of the students have a GPA below the graduation requirement of 3.00?

Suppose we have a random sample of 50 ENM students. Assuming their GPAs are independent, what is the probability that at least 5 of them will have a GPA below 3.00?

Solution

So percentage of the students have a GPA below the graduation requirement of 3.00 is

P(X<3) = P((X-mean)/s <(3-3.2)/0.4)

=P(Z<-0.5) =0.3085 (from standard normal table)

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Given X follows Binomial distribution with n=50 and p=0.3085

P(X=x)=50Cx*(0.3085^x)*((1-0.3085)^(50-x)) for x=0,1,2,..,50

So he probability that at least 5 of them will have a GPA below 3.00 is

P(X>=5)=1-P(X=0)-P(X=1)-...-P(X=4)

=1-50C0*(0.3085^0)*((1-0.3085)^(50-0))-...-50C4*(0.3085^4)*((1-0.3085)^(50-4))

=0.9998913