Question

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

The number of questionnaires to be mailed is approximately 292.

Step-by-step solution

Given information 

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

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,

$\begin{array}{c}\mu =np\\ =n\times 0.4\\ =0.4n\end{array}$

Also, the standard deviation of the binomial distribution is,

$\begin{array}{c}\sigma =\sqrt{npq}\\ =\sqrt{n\times 0.4\times 0.6}\\ =\sqrt{0.24n}\end{array}$

By the Empirical rule, almost 95% of all the values are larger than$\mu -2\sigma$.

Therefore,$\mu -2\sigma >100$

Substitute the values,

$\begin{array}{c}0.4n-2\sqrt{0.24n}>100\\ 0.4n-2\sqrt{0.24n}-100>0\end{array}$

Solve the above equation for $\sqrt{n}$,

$\begin{array}{c}\sqrt{n}=\frac{0.98\pm \sqrt{{0.98}^2-4\times 0.4\times (-100)}}{2\times 0.4}\\ =\frac{0.98\pm \sqrt{0.90604+160}}{0.8}\\ =\frac{0.98\pm 12.687}{0.8}\end{array}$

$\begin{array}{c}=\frac{13.667}{0.8}\\ =17.08375\\ n=291.9\approx 292\end{array}$

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