Mathematics, 02.04.2021 01:00 1234567890lknn

Suppose that a random sample of 20 items is selected from the machine. If the machine produces 5% defectives, find the probability that the sample will contain at least three defectives, by using the following methods. (a) the normal approximation to the binomial (Round your answer to four decimal places.) (b) the exact binomial tables (Round your answer to three decimal places.)

Answers: 2

Show answers

Answers

Answer from: Aj85537

a) 0.0618 = 6.18% probability that the sample will contain at least three defectives.

b) 0.076 = 7.6% probability that the sample will contain at least three defectives

Step-by-step explanation:

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.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$ , $\sigma = \sqrt{V(X)}$ .

Sample of 20 items is selected from the machine.

This means that n = 20

5% defectives

This means that p = 0.05

(a) the normal approximation to the binomial

The mean is:

$\mu = E(X) = np = 20*0.05 = 1$

The standard deviation is:

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{20*0.05*0.95} = 0.9747$

The probability is, using continuity correction, $P(X \geq 3 - 2.5) = P(X \geq 2.5)$ , which is 1 subtracted by the pvalue of Z when X = 2.5. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{2.5 - 1}{0.9747}$

Z = 1.54

Z = 1.54 has a pvalue of 0.9382

1 - 0.9382 = 0.0618

0.0618 = 6.18% probability that the sample will contain at least three defectives.

(b) the exact binomial tables

This is:

$P(X \geq 3) = 1 - P(X < 3)$

In which

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

In which

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

$P(X = 0) = C_{20,0}.(0.05)^{0}.(0.95)^{20} = 0.358$

$P(X = 1) = C_{20,1}.(0.05)^{1}.(0.95)^{19} = 0.377$

$P(X = 2) = C_{20,2}.(0.05)^{2}.(0.95)^{18} = 0.189$

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.358 + 0.377 + 0.189 = 0.924

$P(X \geq 3) = 1 - P(X < 3) = 1 - 0.924 = 0.076$

0.076 = 7.6% probability that the sample will contain at least three defectives

Answer from: Quest

is means division by fractions

step-by-step explanation:

Answer from: Quest

Ithink it’s b, hope this .

Another question on Mathematics

Mathematics, 21.06.2019 16:10

What is the point-slope equation of a line with slope -5 that contains the point (6,3)

Answers: 1

Answer

Mathematics, 21.06.2019 19:30

What is the slope of the line shown below?

Answers: 2

Answer

Mathematics, 21.06.2019 22:00

Melanie is buying a bathing suit that costs 39.99. she is given a discount at the cash register of $6.00 when she scratched 15% off on a game piece for a store promotion. what will the sale price of melanie’s bathing suit be?

Answers: 1

Answer

Mathematics, 21.06.2019 22:00

Aprofessional hockey team has noticed that, over the past several seasons, as the number of wins increases, the total number of hot dogs sold at the concession stands increases. the hockey team concludes that winning makes people crave hot dogs. what is wrong with the hockey team's conclusion? a) it does not take into account the attendance during each season. b) it does not take into account the number of hamburgers sold during each season. c) it does not take into account the number of concession stands in the hockey arena. d) it does not take into account the number of seasons the hockey team has been in existence.

Answers: 1

Answer

You know the right answer?

Suppose that a random sample of 20 items is selected from the machine. If the machine produces 5% de...

Questions

Mathematics, 04.03.2022 14:00

3 over 4x + 6 = 33 i need help really bad i dont understand it...