Mathematics, 27.07.2021 22:10 nan487

A company that produces DVD drives has a 12% defective rate. Let X represent the number of defectives in a random sample of 55 of their drives. Required:
a. What is the probability the sample will contain exactly 8 defective drives?
b. What is the probability the sample will contain more than 8 defective drives?
c. What is the probability the sample will contain less than 8 defective drives?
d. What is the expected number of defective drives in the sample?

Answers: 1

Show answers

Answers

Answer from: robert7248

a) 0.1287 = 12.87% probability the sample will contain exactly 8 defective drives

b) 0.2092 = 20.92% probability the sample will contain more than 8 defective drives.

c) 0.6621 = 66.21% probability the sample will contain less than 8 defective drives.

d) The expected number of defective drives in the sample is 6.6

Step-by-step explanation:

For each DVD, there are only two possible outcomes. Either it is defective, or it is not. The probability of a DVD being defective is independent of any other DVD, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

A company that produces DVD drives has a 12% defective rate.

This means that p = 0.12

Let X represent the number of defectives in a random sample of 55 of their drives.

This means that n = 55

a. What is the probability the sample will contain exactly 8 defective drives?

This is P(X = 8) . So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 8) = C_{55,8}.(0.12)^{8}.(0.88)^{47} = 0.1287$

0.1287 = 12.87% probability the sample will contain exactly 8 defective drives.

b. What is the probability the sample will contain more than 8 defective drives?

This is:

$P(X 8) = 1 - P(X \leq 8)$

In which:

$P(X \leq 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)$

Then

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{55,0}.(0.12)^{0}.(0.88)^{55} = 0.0009$

$P(X = 1) = C_{55,1}.(0.12)^{1}.(0.88)^{54} = 0.0066$

$P(X = 2) = C_{55,2}.(0.12)^{2}.(0.88)^{53} = 0.0244$

$P(X = 3) = C_{55,3}.(0.12)^{3}.(0.88)^{52} = 0.0588$

$P(X = 4) = C_{55,4}.(0.12)^{4}.(0.88)^{51} = 0.1043$

$P(X = 5) = C_{55,5}.(0.12)^{5}.(0.88)^{50} = 0.1450$

$P(X = 6) = C_{55,8}.(0.12)^{6}.(0.88)^{49} = 0.1648$

$P(X = 7) = C_{55,7}.(0.12)^{7}.(0.88)^{48} = 0.1573$

$P(X = 8) = C_{55,8}.(0.12)^{8}.(0.88)^{47} = 0.1287$

$P(X \leq 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 0.0009 + 0.0066 + 0.0244 + 0.0588 + 0.1043 + 0.1450 + 0.1648 + 0.1573 + 0.1287 = 0.7908$

$P(X 8) = 1 - P(X \leq 8) = 1 - 0.7908 = 0.2092$

0.2092 = 20.92% probability the sample will contain more than 8 defective drives.

c. What is the probability the sample will contain less than 8 defective drives?

This is:

P(X < 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

With the values we found in b.

P(X < 8) = 0.0009 + 0.0066 + 0.0244 + 0.0588 + 0.1043 + 0.1450 + 0.1648 + 0.1573 = 0.6621

0.6621 = 66.21% probability the sample will contain less than 8 defective drives.

d. What is the expected number of defective drives in the sample?

The expected value of the binomial distribution is:

E(X) = np

In this question:

E(X) = 55(0.12) = 6.6

The expected number of defective drives in the sample is 6.6

Answer from: Quest

abde or 1,2,4, 5 is the answer

step-by-step explanation:

Which statements must be true about the reflection of δxyz across ? check all that apply. m∠x'z'y'

Answer from: Quest

lo.lll

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 19:30

The amount spent on food and drink by 20 customers is shown below what is the modal amount spent and the median? 40 25 8 36 18 15 24 27 43 9 12 19 22 34 31 20 26 19 17 28

Answers: 1

Answer

Mathematics, 21.06.2019 21:30

Joe has a rectangular chicken coop. the length of the coop is 4 feet less than twice the width. the area of the chicken coop is 510 square feet. what is the length of the chicken coup?

Answers: 3

Answer

Mathematics, 21.06.2019 22:30

The dash figure is a dilation if the original figure. find the scale factor. classify the dilation as an enlargement or a reduction

Answers: 2

Answer

Mathematics, 21.06.2019 23:30

The approximate line of best fit for the given data points, y = −2x − 2, is shown on the graph.which is the residual value when x = –2? a)–2 b)–1 c)1 d)2

Answers: 2

Answer

You know the right answer?

A company that produces DVD drives has a 12% defective rate. Let X represent the number of defective...

Questions