When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let x denote the number of defective boards in a batch of 25 boards, so that x has a binomial distribution with n 5 25 and 5 .05. D

Question:

When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let x denote the number of defective boards in a batch of 25 boards, so that x has a binomial distribution with n = 25 and π = 0.05.

(a)What proportion of batches have at most 2 defective boards?

(b)What proportion of batches have at least 5 defective boards?

(c)What proportion of batches will have all 25 boards free of defects?

(a) Proportion of batches have at most 2 defective boards 0.87291.

(b) Proportion of batches have at least 5 defective boards is 0.00714.

(c) Proportion of batches will have all 25 boards free of defects is 0.2774.

Solution:

n= 25,  π = 0.05

Binomial distribution formula:

(a) Proportion with x = 0 is given by

P(0) = 0.2774

Proportion with x = 1 is given by

P(1) = 0.36499

Proportion with x = 2 is given by

P(2) = 0.23052

Proportion with at most 2 defective = P(x ≤ 2)

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

                                                              = 0.2774 + 0.36499 + 0.23052

Proportion with at most 2 defective = 0.87291

(b) Proportion with x = 3 is given by

P(3) = 0.093

Proportion with x = 4 is given by

P(4) = 0.2693

Proportion with x ≥ 5 = 1 – P(x ≤ 5)

                                   = 1 – (P(0) + P(1) + P(2) + P(3) + P(4))

                                   = 1 – (0.2774 + 0.36499 + 0.23052 + 0.093 + 0.093)

                                   = 0.00714

Proportion of batches have at least 5 defective boards is 0.00714.

(c) Proportion with x = 0

It is already found in (a).

Proportion of batches will have all 25 boards free of defects is 0.2774.