Question

Pleas TYPE your answers to the following questions. Thank you!

A random set of 24 products is sampled from a production process. The fraction of non-conforming products manufactured by the process is thought to be 0:1, and each product is independently defective of the others.

(a) What is the probability of 0 nonconforming products in the sample? 6 nonconforming products? 12 nonconforming products? 24 nonconforming products?

(b) What is the probability that at most 4 products won't conform to specifications?

(c) What are the expectation and standard deviation of the number of products that won't conform to specification in such a random sample?

(d) Simulate and provide 20 independent samples from this production process using JMP. What are the sample mean and standard deviation across your samples?

(e) Explain the gap (if any) between your answers to the previous two questions.

Answer 1

The binominal distribution formula is given by

$P(x) = \frac{n!}{(n-x)!x!}p^xq^{n-x}$

where n the total number of trails
x is the number of successes out of n trials
p is the probability of success
q is the probability of failure

In this case we have

n = 24
p = 0.1
q = 0.9

What is the probability of 0 nonconforming products in the sample?
$\\X= 0\\ P(x) = \frac{n!}{(n-x)!x!}p^x\times q^{n-x}\\ P(0) = \frac{24!}{(24-0)!}\times (0.1)^{0} \times (0.9)^{24-0} = 0.0798\\$

What is the probability of 6 nonconforming products in the sample?
$\\X= 6\\ P(x) = \frac{n!}{(n-x)!x!}p^x\times q^{n-x}\\ P(6) = \frac{24!}{(24-6)!}\times (0.1)^{6} \times (0.9)^{24-6} = 0.0202\\$

What is the probability of 12 nonconforming products in the sample?
$\\X= 12\\ P(x) = \frac{n!}{(n-x)!x!}p^x\times q^{n-x}\\ P(12) = \frac{24!}{(24-12)!}\times (0.1)^{12} \times (0.9)^{24-12} = 0.0000\\$

What is the probability of 24 nonconforming products in the sample?
$\\X= 24\\ P(x) = \frac{n!}{(n-x)!x!}p^x\times q^{n-x}\\ P(12) = \frac{24!}{(24-24)!}\times (0.1)^{24} \times (0.9)^{24-24} = 0.0000\\$

What is the probability that at most 4 products won't conform to specifications?

$P(X\le4) = P(0)+P(1)+P(2)+P(3)+P(4)=0.9149$

What are the expectation and standard deviation of the number of products that won't conform to specification in such a random sample?

$\\E(X) = n \times p = 24 \times 0.1 = 2.4\\ \sigma = \sqrt{n \times p \times q}= \sqrt{24 \times 0.1 \times 0.9}= 1.4696$