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

i think it is x = 1.2

Answer from: Quest

answer: get more recipes with 3 magazines for $20! i often use chicken broth instead of water. 3 tablespoons butter, divided; 1/2 cup freshly grated parmigiano-reggiano bring water and salt to a boil in a large saucepan; pour polenta slowly into 5 minutes to thicken; stir and taste for salt before transferring to a serving bowl.

Another question on Mathematics

Mathematics, 21.06.2019 13:00

Which expression is equivalent to −16y+13 ? −16(−y+13) 16(−y+2) −16(y+2) 16(−y+13)

Answers: 1

Answer

Mathematics, 21.06.2019 13:30

Write the converse of the following statement: if the trees have no leaves, then it is fall. if the trees have no leaves, then it is fall. the trees have no leaves, therefore it is fall. it is fall since the trees have no leaves. if it is fall, then the trees have no leaves.

Answers: 2

Answer

Mathematics, 21.06.2019 16:00

Arectangle with an area of 47 m² is dilated by a factor of 7. what is the area of the dilated rectangle? enter your answer in the box. do not leave your answer as a fraction.

Answers: 1

Answer

Mathematics, 21.06.2019 17:40

A. 127 cm^2 b. 144.5 cm^2 c. 172 cm^2 d. 50 cm^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

Mathematics, 04.07.2020 14:01

PLEASE HELP, SOLVE THIS PROBLEM AND GIVE ME THE ANSWER!!!

Mathematics, 04.07.2020 14:01