Question

Liz is considering a cellular phone service plan. Under this plan, she would specify a quantity of minutes, say x, per month that she would buy at 5¢ per minute. Hence, 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. ) Liz 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. How many minutes should she contract for?

For each unused minute at the end of a month, how much does it cost Liz?

$0.45

$0.05

None of the above

$0.40

$0.35

What is the probability that Liz’s monthly usage is less than 250 minutes?

75%

50%

Less than 40%

60%

More than 80%

How many minutes should she contract for per month to minimize her expected total monthly cost?

304 minutes

263 minutes

278 minutes

271 minutes

274 minutes

If the price of an extra minute increases to 50¢ per minute, how should Liz adjust the amount of minutes she contracts for to minimize her expected total monthly cost?

It cannot be determined

None of the above

She should contract for more minutes

She should contract for the same amount of minutes

She should contract for less minutes

When should Liz double the minutes she contracts for to minimize her expected total monthly cost?

The price of a minute in the contract doubles to 10¢ per minute

The mean of her monthly needs doubles to 500 minutes

None of the above

The standard deviation of her monthly needs doubles to 48 minutes

The price of an extra minute doubles to 80¢ per minute

Answer 1

a) It costs Liz 0.05c per minute. She cannot recover the unused minutes charged at this rate.

b) Mean, u = 250 minutes, Std deviation,s = 24 minutes

X = 250 minutes

Z = (X-u) /s = (250-250) / 24 = 0.00

For Z = 0, Probabilty = 50% (Standard normal distribution table)

c) Cost of having extra minutes than required, Co = 0.05

Cost of having lower minutes than required , Cu = 0.40

Critical ratio factor = Cu / Cu+Co = 0.88

For p = 0.8889, Z -value = 1.18

Optimal number of mintues = Mean + Std deviation * Z score = 250+24*1.18 = ~ 278 minutes

d) Critical ratio if price of extra minute is 50c = 0.50 / (0.50+0.05) = 0.90

As critical ratio increases, Z value increases. Hence,  She should contract for more minutes

e) The mean of her monthly need doubles. All the other factors impact critical ratio and Z-score. However, this will lead to a small change and not impact so much that Liz requires 2X minutes. However, when the mean demand itself doubles, she must consider doubling the minutes.