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Chapter 4: Q201SE (page 286)

How many questionnaires to mail? The probability that a consumer responds to a marketing department’s mailed questionnaire is 0.4. How many questionnaires should be mailed if you want to be reasonably certain that at least 100 will be returned?

Short Answer

Expert verified

The number of questionnaires to be mailed is approximately 292.

Step by step solution

01

Given information 

The probability that a consumer responds to a marketing department’s mailed questionnaire is 0.4.

02

Calculating the number of questionnaires 

Define the random variable x as the consumer who responds which follows binomial distribution.

Since, the mean of the binomial distribution is,

μ=np=n×0.4=0.4n

Also, the standard deviation of the binomial distribution is,

σ=npq=n×0.4×0.6=0.24n

By the Empirical rule, almost 95% of all the values are larger thanμ−2σ.

Therefore,μ−2σ>100

Substitute the values,

0.4n−20.24n>1000.4n−20.24n−100>0

Solve the above equation for n,

n=0.98±0.982−4×0.4×(−100)2×0.4=0.98±0.90604+1600.8=0.98±12.6870.8

=13.6670.8=17.08375n=291.9≈292

Thus, the number of questionnaires to be mailed is approximately 292.

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