Chapter 18, Problem 25E

To determine

To explain how many seats should be in the non-smoking area in order to be very sure of having enough seating there and comment on the assumptions and conditions that support your model and explain what “very sure” means to you.

Answer to Problem 25E

The restaurant needs at least $89$ seats in the non-smoking section.

Explanation of Solution

It is given in the question that about $60\%$ of the customers demand a smoke-free area and a new restaurant with $120$ seats is being planned. Thus, let us first verify the assumptions and conditions for the analysis as:

Random condition: It is satisfied as we assume that the customers were randomly selected.

  $10\%$ condition: The sample size less than $10\%$ of the population size.

Success/failure condition: as $np=72,n(1-p)=48$ and both are greater than ten then this is also true.

Thus, all the conditions and assumptions are met. Thus, now find the mean and the standard deviation for the following as:

  $\begin{array}{l}\mu =p=0.60\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.60(0.40)}{120}}\\ =0.045\end{array}$

Thus, for calculating the seats that should be in the non-smoking area in order to be very sure of having enough seating there we will use three standard deviations above the expected proportion and this allow us to be very sure about this. Thus, we have,

  $\begin{array}{l}\mu +3\sigma =0.60+3(0.045)\\ =0.60+0.135\\ =0.735\end{array}$

As the total was $120$ seats being planned, so, we have,

  $\begin{array}{l}120\times 0.735=88.2\\ \approx 89\end{array}$

Thus, the restaurant needs at least $89$ seats in the non-smoking section.