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# Among homeowners in a metropolitan area, 75% recycle

ISBN: 9781118539712 55

## Solution for problem 109E Chapter 4.7

Applied Statistics and Probability for Engineers | 6th Edition

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Applied Statistics and Probability for Engineers | 6th Edition

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Problem 109E

Among homeowners in a metropolitan area, 75% recycle plastic bottles each week. A waste management company services 1500 homeowners (assumed independent). Approximate the following probabilities:

(a) At least 1150 recycle plastic bottles in a week

(b) Between 1075 and 1175 recycle plastic bottles in a week

Step-by-Step Solution:

Step 1 of 2</p>

(a)

Among homeowners in a metropolitan area,  recycled plastic bottles each week.

A waste management company services  homeowners (assumed independent).

We are asked to find the probability that the at least  recycle plastic bottles in a week.

Let  denotes the recycle plastic bottles in a week.

Since homeowners independently recycle plastic bottles each week with the probability

Hence  follows a binomial distribution with parameter

We need to find the probability

We will use approximation here which can be defined as,

If  is a binomial random variable with parameters  and ,

Is approximately a standard normal random variable.

To approximate a binomial probability with a normal distribution, a continuity correction is applied as follows:

The approximation is good for

Since  we can use the approximation.

Hence the probability  we can write,

We can write  as,

From the z table, the area to the left of  is .

Hence the probability that the at least  recycle plastic bottles in a week is

Step 2 of 2

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