Question

Sarah is planning her wedding. She and her fiancé have signed a contract with a caterer that calls for them to tell the caterer the number of guests that will attend the reception a week before the actual event. This "final number" will determine how much they will have to pay the caterer; they must pay $60 per guest that they commit to. For example, if they tell the caterer that they expect 200 guests, they must pay the caterer $12,000 (=200 × 60) even if, say, 190 guests show up. The contract calls for a higher rate of $85 per extra guest for the number of guests beyond what the couple commits to. Thus, if Sarah and her fiancé commit to 200 guests but 203 show up, they must pay $12,255 (the original 12,000 plus 3 × 85). The problem Sarah faces is that she still does not know the exact number of guests to expect. Despite asking that friends and family members reply to their invitations a month ago, some uncertainty remains: e.g., her brother may or may not bring his new girlfriend. Sarah has determined that the expected number of guests (i.e., the mean number) is 200, but the actual number could be anywhere from 168 to 232. [A normal distribution with mean 200 and standard deviation 10 seems a good approximation for the number of guests.]A. How many guests should Sarah commit to with the caterer?

Answer 1

Sarah should commit to with the caterer 195 guests

Step-by-step explanation:

1. Let's review the information provided to us to answer the question correctly:

Overage cost = $ 60 (This is the cost incurred per guest that doesn't show up)

Underage cost = $ 25 (85 - 60) (This is the cost incurred per extra guest beyond what the couple commits to)

Expected number of guests or mean of the normal distribution = 200

Standard deviation = 10

2. How many guests should Sarah commit to with the caterer?

For solving this question, we should calculate the critical ratio, this way:

F(Q) = 25/(60 + 25)

F(Q) = 25/85 = 0.2941

This number is the probability of a z-score less than the z-score of the mean (0.5). Using our z-table, we find out that:

P (z ≤ -0.54) = 0.2946

Using this z-score and the standard deviation we calculate the number of guests Sarah should commit with the caterer, this way:

-0.54 = (x - 200)/10

-5.4 = x - 200

x = 200- 5.4

x = 194.6

x = 195 (rounding to the next whole)

Answer 2

204 guest

Step-by-step explanation:

The overage cost is $60 because ordering a plate for a guest that doesn’t show up costs $60. The underage cost is = $85 – 60 = $25, because not committing to a guest that does show up costs an extra $25. The critical ratio is 25/(60 + 25) = 0.2941. From the table, and , so the optimal number of guests to commit to is 204.