(10+ 10+ 10 Points] Phoenix water is provided to approximately 1.4 million peopl

Question

(10+ 10+ 10 Points] Phoenix water is provided to approximately 1.4 million people who are served through more than 362,000 accounts (http://phoenix.gov/WATER/wtrfacts.html). All accounts are metered and billed monthly. The probability that an account has an error in a month i be assumed to be independent. (a) What are the mean and standard deviation of the number of account errors each month? (b) Find the probability of fewer than 345 errors in a month. (c) Find a value so that the probability that the number of errors exceeds this value is 0.10. 3.

Explanation / Answer

(a)

Let Number of accounts be n and Probability of account in error be p.

Then n = 362000 and p = 0.001

Assuming that the number of accounts error follow binomial distribution,

Mean number of account errors = n * p = 362000 * 0.001 = 362

Variance of number of account errors = n p (1-p)

= 362000 * 0.001 * (1 - 0.001) = 361.638

Standard deviation of number of account errors = sqrt(361.638) = 19.02

(b)

As, np > 5 and n(1-p) > 5, we can approximate the binomial distribution of number of accounts error as a normal distribution with mean 362 and standard deviation of 19.02

Probability of fewer than 345 errors in a month = P(X < 345)

= P(Z < (345-362)/ 19.02)

= P(Z < -0.8938)

= 0.1857 (Using Z table)

Probability of fewer than 345 errors in a month is 0.1857

(c)

Let x be the value that the probability that the number of errors exceeds this value is 0.10. So,

P(Z > z) = 0.1

Using Z table, we get z = 1.28

Then,

1.28 = (x - 362) / 19.02

=> x - 362 = 1.28 * 19.02 = 24.35

=> x = 362 + 24.35 = 386.35

So, 386.35 is the value such that the probability that the number of errors exceeds this value is 0.10

Related Questions

Navigate

• Browse (All)
• Browse #
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.