[10+ 10+ 10 Points Phoenix water is provided to approximately 1.4 million people

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 is 0.001, and accounts can 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

Solution:

Part a

Mean = n*p

Standard deviation = sqrt(n*p*q) where q = 1 – p

We are given,

n = 362000

p = 0.001

q = 1 – 0.001 = 0.999

Mean = 362000*0.001 = 362

Standard deviation = sqrt(362000*0.001*0.999) = 19.01678

Part b

We have to find P(X<345)

By using continuity correction, we have to find P(X<345 – 0.5) = P(X<344.5)

Z = (X – mean)/SD

Z = (344.5 - 362) / 19.01678

Z = -0.92024

P(Z<-0.92024) = P(X<345) = 0.178724

Required probability = 0.178724

Part c

We are given

P(X>x) = 0.10

So, P(X<x) = 0.90

So, critical Z value = 1.281552

(By using z-table or excel)

X = Mean + Z*SD

X = 362 + 1.281552*19.01678

X = 386.371

Required value = 386.371

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