The ideal size of a class of a college is 150.

Question

The ideal size of a class of a college is $150$ students. The college, experienced from past, knows that only $30$% of the admitted students will actually attend. The college uses a policy of approving the application of $450$ students. Find the probability that more than $150$ students attend the class. (Given area under the standard normal curve enclosed between the ordinates z=$0$ and z= $1.59$ is $0.441$.

Ans : 0.0559

Now let X be the random variable indicating the number of students present. Now X follows binomial distribution with parameters (n,p)=(450,½)

Now as n -> infinity and p is finite so we use binomial approximation to normal distribution.

np=450*½=225 ; √(np(1-q))=√(225/2)

We are required to find,

P(X>150)=P(X>150.5) [for discrete to continuous approximation] =P(Z>(150.5-225)/√(225/2)) =P(Z>-7.023) And this value is not even available in any standard normal table.

Answer 1

Now let X be the random variable indicating the number of students present.

Now it is given that the probability of a student being present is $30$% so,

Now X follows binomial distribution with parameters (n,p)=(450,0.3)

Now as n -> infinity and p is finite so we use binomial approximation to normal distribution.

np=450*0.3=135 ;

√(np(1-p))=√(135*0.7)

We are required to find,

P(X>150)=P(X>150.5) [for discrete to continuous approximation]

=P(Z>(150.5-135)/√(135*0.7))

=P(Z>1.594)

=1-P(Z<1.59)

=1-(0.5+0.441)

=0.059