Chapter 18, Problem 26E

To determine

To explain how many of those meals should he plan on serving in order to be pretty sure of having enough steaks on hand to meet customer demand and justify your answer including an explanation of what “pretty sure” means to you.

Answer to Problem 26E

The chef will need at least $47$ steaks on hand.

Explanation of Solution

It is given in the question that a restaurant serving about $180$ people of Friday evening and believes that about $20\%$ of the patrons will order the chef’s steak special. 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 $np=36,n(1-p)=144$ 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:

  $\begin{array}{l}\mu =p=0.20\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.20(0.80)}{180}}\\ =0.030\end{array}$

Thus, for calculating those meals should he plan on serving in order to be pretty sure of having enough steaks on hand to meet customer demand we will use two standard deviations above the expected proportion and this allow us to be pretty sure about this. Thus, we have,

  $\begin{array}{l}\mu +2\sigma =0.20+2(0.030)\\ =0.20+0.060\\ =0.2596\end{array}$

As the total was $180$ people of Friday evening, so, we have,

  $\begin{array}{l}180\times 0.2596=46.733\\ \approx 47\end{array}$

Thus, the chef will need at least $47$ steaks on hand.