A company determines that one out of 25 of the pens it produces is defective. A

Question

A company determines that one out of 25 of the pens it produces is defective. A random sample of 30 pens tested and the number of defective pens is counted. The number of defective pens out of 30 fits a ______ situation. Define the random variable: X =______ Number of trials: n =_____ Success for this situation is______ Probability of success: p =______Probability of failure: q = _____ Write each of the following as a probability, i.e. P(X lessthanorequalto 2), and find the probability: a) Find the probability that none of the pens is defective. b) Find the probability that three of the pens are defective. c) Find the probability that less than two of the pens are defective. d) Find the probability that the number of defective pens is at least ten. e) Find the probability that between three and five of the pens are defective. f) Find the mean and standard deviation for the number of defective pens in a sample of 30 pens.

Explanation / Answer

Number of defective pens out of 30 fits Binomial distribution

Random variable x: Number of defective pens out of 30 pens.

Rang of x: 0, 1, 2, 3, …………………………..,30

Number of trials n = 30

Success of this situation is, if we get defective pen then we say success is occur.

Probability of success P = 1/25 = 0.04

Probability of failure : ( 1 – P ) = 0.96

A)

P (none of pen is defective) = P (x = 0)

Using Excel function,

=BINOMDIST(Number_s , trials, probability of success, cumulative )

Plug cumulative is 0 if you have to find P (X = x) and 1 if you have to find P (X x)

=BINOMDIST( 0, 30, 0.04, 0 ) = 0.2938

P (X = 0 ) = 0.2939

B)

P (three pens are defective) = P ( X = 3 )

= BINOMDIST( 3,30,0.04,0) = 0.0863

C)

P( less than 2 pens are defective ) = P( X< 2 ) = P(X 1 )

= BINOMDIST( 1,30,0.04,1) = 0.6612

D) P( at least 10 pens are defective ) = P( X 10 ) = 1 – P( X < 10 ) = 1 – P( X 9 )

= 1 -   BINOMDIST( 9,30,0.04,1) = 1 – 1 = 0

E)

P( between 3 and 5 pens are defective ) = P( 3 X 5 ) = P( X 5) - P( X 3)    

=BINOMDIST(5,30,0.04,1) - BINOMDIST(3,30,0.04,1)

=0.998939 – 0.969407 = 0.02953

F)

Mean = n * P = 30*0.04 = 1.2

standard deviation = square root of ( n*P*( 1- P )) = 1.0733

                                                                                                                                                                                           

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