$($Sarah$$s Wedding$)$ Sarah is planning her wedding. She and her fiance$$ 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 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\backslash$times $$60)$ even if only, say, $84$ guests show up$.$ The contract calls for a higher rate of $$85$ per extra guest for the number of guests beyond what the couple commits to$.$ Thus, if Sarah and her fiance$$ commit to $90$ guests but $92$ show up$,$ they must pay $$5,570($the original $$5,400$ plus $2\backslash$times $$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 fiance$$s college roommate may$$or may not$$be able to take a vacation from work; and so forth. Sarah has determined that the expected number of $334$ Chapter $15$ guests $($i$.$e$.,$ the mean number$)$ is $100,$ but the actual number could be anywhere from $84$ to $116$: a$.$ How many guests should Sarah commit to with the caterer?$[15\mathrm{.}2]$ b$.$ Suppose Sarah commits to $105$ guests. What is Sarah$$s expected bill?$[15\mathrm{.}2]$ 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.$$ For example, if they commit to $90$ but only $84$ show, they will have to pay $84\backslash$times $$60=6\backslash$times $$45=$ $$5,310.$ Now how many guests should she commit to$?[15\mathrm{.}4]$ 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$)?[15\mathrm{.}4$