When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let X = the number of defective boards in a random sample of size n = 25, so X ~ Bin(25, 0.05). (Round your probabilities to three decimal places.)(a) Determine P(X ≤ 2).(b) Determine P(X ≥ 5).(c) Determine P(1 ≤ X ≤ 4).(d) What is the probability that none of the 25 boards is defective?(e) Calculate the expected value and standard deviation of X. (Round your standard deviation to two decimal places.)

a) P(X ≤ 2) = 0.87

b)  P(X ≥ 5) = 0.01

c) P(1 ≤ X ≤ 4) = 0.71

d) P ( X = 0 ) = 0.28

e) σ(X) = 1.09 , E(X) = 1.25

Step-by-step explanation:

Given:

- Let X = the number of defective boards in a random sample of size n = 25, so X ~ Bin(25, 0.05)

Where, n = 25 and p = 0.05

Find:

(a) Determine P(X ≤ 2).(b) Determine P(X ≥ 5).(c) Determine P(1 ≤ X ≤ 4).(d) What is the probability that none of the 25 boards is defective?(e) Calculate the expected value and standard deviation of X.

Solution:

- The probability mass function for a binomial distribution is given by:

                         P ( X = x ) = nCr * (p)^r * ( 1 - p )^(n-r)

a) P(X ≤ 2):

                         P(X ≤ 2) = P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 )

                         = (0.95)^25 + 25*0.05*0.95^24 + 25C2*0.05^2*0.95^23

                         = 0.87

b) P(X ≥ 5):

            P(X ≥ 5) = 1 - [P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 ) + P ( X = 3 ) + P(X=4)

            = 1 - [ 0.87 + 25C3*0.05^3*0.95^22 + 25C4*0.05^4*0.95^21]

            = 1 - 0.98994

            = 0.01

c) P(1 ≤ X ≤ 4):

            P(1 ≤ X ≤ 4) = P ( X = 1 ) + P ( X = 2 ) + P ( X = 3 ) + P(X=4)

            = ( 0.87 - 0.95^25) + 0.11994

           = 0.71

d) P( X = 0 )

            P ( X = 0 ) = 0.95^25 = 0.28

e) E(X) & σ(X):

            E(X) = n*p

            E(X) = 25*0.05 = 1.25

            σ(X) = sqrt ( Var (X) )

            σ(X) = sqrt ( n*p*(1-p) ) = sqrt ( 25*0.05*0.95 )

            σ(X) = 1.09

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step-by-step explanation:

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both graphs increase on their domain

neither graph has a y-intercept

step-by-step explanation:

the following points can be visualized easily from the graph