Question

In a population of students, 60% study for every homework assignment. A sample of 40 students from this population is taken, and the number who study for every homework assignment from this sample is recorded as the random variable X.

a) (4 pts) Verify that X has a binomial distribution.

b) (4 pts) Find the probability that exactly 25 students from the sample study for the exam.

c) (4 pts) Find the mean for this binomial experiment

Answer 1

X= no. of students who study for every homework assignment

a) X has a binomial distribution because :

i) Each trial is similar and no. of trials are constant

   (ii) There are only two possible outcomes for each trial

(iii) outcome of each trial is independent of other trials and Probability of Success for each trial is constant

no of trials = n= 40

Probability of success( we consider student who study for every homework assignment as success) = p= 60% = 0.6

So X ~ B(40 , 0.6)

b) Probability distribution for a Binomial distribution is given by :

   P(X=x) gives us probability of getting 'x' no. of students who study for every homework assignment out of 40

( is the combination operator)

So probability that Exactly 25 students who study for every homework assignment = P(X=25 )

  

c) Mean of a binomial distribution is given as :

   Mean = E(X)

  

   Mean = n*p = 40 * 0.6 = 24 students