Mathematics, 13.02.2020 02:41 laytonlutz

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

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Answer from: hernandez09297

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.

Solution:

n= 25, π = 0.05

Binomial distribution formula:

$$P(x)=\frac{n !}{x !(n-x) !} \pi^{x}(1-\pi)^{n-x}$

$P(x)=\frac{25 !}{x !(25-x) !} \times(0.05)^{x}(0.95)^{25-x}$

(a) Proportion with x = 0 is given by

$P(0)=\frac{25 !}{0 ! 24 !}(0.05)^0 (0.95)^{25}$

P(0) = 0.2774

Proportion with x = 1 is given by

$P(1)=\frac{25 !}{1 ! 24 !}(0.05)^1 (0.95)^{24}$

P(1) = 0.36499

Proportion with x = 2 is given by

$P(2)=\frac{25 !}{2 ! 23 !}(0.05)^{2}(0.95)^{23}$

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

$P(3)=\frac{25 !}{3 ! 22 !}(0.05)^{3}(0.95)^{22}$

P(3) = 0.093

Proportion with x = 4 is given by

$P(4)=\frac{25 !}{4 ! 21 !}(0.05)^{4}(0.95)^{21}$

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.

It is already found in (a).

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

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Answer from: Quest

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