Marisol is new to town and is in the market for cellular phone service. She has settled on Wildcat Cellular, which will give her a free phone if she signs a one-year contract. Wildcat offers several calling plans. One plan that she is considering is called “Pick Your Minutes.” Under this plan, she would specify a quantity of minutes, say x, per month that she would buy at 5¢ per minute. Thus, her upfront cost would be $0.05x. If her usage is less than this quantity x in a given month, she loses the minutes. If her usage in a month exceeds this quantity x, she would have to pay 40¢ for each extra minute (that is, each minute used beyond x). For example, if she contracts for x = 120 minutes per month and her actual usage is 40 minutes, her total bill is $120 × 0.05 = $6.00. However, if actual usage is 130 minutes, her total bill would be $120 × 0.05 = (130 − 120) × 0.40 = $10.00. The same rates apply whether the call is local or long distance. Once she signs the contract, she cannot change the number of minutes specified for a year. Marisol estimates that her monthly needs are best approximated by the normal distribution, with a mean of 250 minutes and a standard deviation of 24 minutes.
a. If Marisol chooses the “Pick Your Minutes” plan described above, how many minutes should she contract for? [15.4]
b. Instead, Marisol chooses to contract for 240 minutes. Under this contract, how much (in dollars) would she expect to pay at 40 cents per minute? [15.4]
c. A friend advises Marisol to contract for 280 minutes to ensure limited surcharge payments (i.e., the 40-cents-per-minute payments). Under this contract, how many minutes would she expect to waste (i.e., unused minutes per month)? [15.4]
d. If Marisol contracts for 260 minutes, what would be her approximate expected monthly cell phone bill? [15.4]
e. Marisol has decided that she indeed does not like surcharge fees (the 40-cents-per-minute fee for her usage in excess of her monthly contracted minutes). How many minutes should she contract for if she wants only a 5 percent chance of incurring any surcharge fee? [15.4]
f. Wildcat Cellular offers another plan called “No Minimum” that also has a $5.00 fixed fee per month but requires no commitment in terms of the number of minutes per month. Instead, the user is billed 7¢ per minute for her actual usage. Thus, if her actual usage is 40 minutes in a month, her bill would be $5.00 = 40 × 0.07 = $7.80. Marisol is trying to decide between the “Pick Your Minutes” plan described above and the “No Minimum” plan. Which should she choose?