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

Answer

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(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