Sarah is planning her wedding She and her fianc have
Sarah is planning her wedding. She and her fiancé have signed a con- tract 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 have to pay the caterer; they must pay \$60 per guest that they commit to. If, for example, they tell the caterer that they expect 90 guests, they must pay \$5,400 (90 \$60) even if only, say, 84 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 90 guests but 92 show up, they must pay \$5,570 (the original \$5,400 plus 2 \$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: her brother may--or may not--bring his new girlfriend; her fiancé's college roommate may--or may not--be able to take a vacation from work; and so forth. Sarah has determined that the expected number of guests (i.e., the mean number) is 100, but the actual number could be anywhere from 84 to 116:
a. How many guests should Sarah commit to with the caterer?
b. Suppose Sarah commits to 105 guests. What is Sarah's expected bill?
c. Suppose that the caterer is willing to alter the contract so that if fewer than the number of guests they commit to show up, they will get a partial refund. In particular, they only have to pay \$45 for each "no-show." For example, if they commit to 90 but only 84 show, they will have to pay 84 \$60 6 \$45 \$5,310. Now how many guests should she commit to?
d. The caterer offers Sarah another option. She could pay \$70 per guest, no matter how many guests show up; that is, she wouldn't have to commit to any number before the wedding. Should Sarah prefer this option or the original option (\$60 per committed guest and \$85 each guest beyond the commitment)?
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