# Question

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 $0.05 x. 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?

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?

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)?

d. If Marisol contracts for 260 minutes, what would be her approximate expected monthly cell phone bill?

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?

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?

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?

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?

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)?

d. If Marisol contracts for 260 minutes, what would be her approximate expected monthly cell phone bill?

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?

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?

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