## Question

1) (2.5 points) A small business receives a daily income that is normally distributed with mean $1600 and standard deviation $300. There are two daily costs – one that depends on the daily income and one that does not. The first cost is normally distributed with a mean equal to one half of the random daily income and with a standard deviation equal to one tenth of the random daily income. The second cost is normally distributed with mean $550 and standard deviation $150. Using Excel, simulate the daily profit (income – costs) for this business numerous (several thousand) times. Estimate a) the average daily profit, b) the proportion of days that result in a negative profit (a loss), c) the proportion of days with profit greater than $400 and d) the profit that represents the top 5% of outcomes (use =percentile).

3) (2.5 points) A credit union has recently installed a new, remote ATM in downtown Durango. They must decide how much cash to fill it with each week. The ATM will be available 12 hours per day. They anticipate that every hour, it is equally likely that 0, 1, 2, 3, or 4 customers will use the ATM. Each customer will be able to withdraw $20, $40, $100, or $200. The probabilities that a customer will withdraw these amounts are, respectively, 0.3, 0.4, 0.2, and 0.1.

Using Excel, simulate withdrawals for one week (84 hours). Run your weekly simulation 25 times (using the F9 function key to generate new data). From these 25 trials, estimate a) the average cash withdrawn from the ATM per week and b) the probability that the ATM will require more than $12,500 per week (the maximum amount of cash that the ATM can hold) to meet customer demand.

4) (2.5 points) A saleswoman must choose the correct cellular phone calling plan that will minimize her average cost per month. She typically makes about 7 hours worth of calls per month, but this number is highly variable, approximately ± 2 hours per month. In fact, assume that the amount of time that she spends on the phone is normally distributed with mean 7 hours and standard deviation 2 hours.

There are several calling plans available. For example, the saleswoman can get 300 minutes per month for a base cost of $30, but every additional minute over the 300 limit costs $0.50. On the other extreme, she can get 700 minutes for $70, with each overage minute costing $0.10. All the choices are listed in the table.

Plan Mins Base Overage

A 300 $30 $0.50/min

B 400 $40 $0.40/min

C 500 $50 $0.30/min

D 600 $60 $0.20/min

E 700 $70 $0.10/min

Which calling plan will minimize her average monthly cost and what will the average cost be? Use simulation to estimate the average cost of each plan and choose the best one. List all of the costs on the cover sheet.

5) (1 point) For Simulation Example A) (available on the course web page), what percentage of the time are the four potential critical paths (ACGJ, ADHJ, BEHJ, and BFI) actually critical?

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