X 2. A hotel near the university always fils up on the evening before football g

Question

X 2. A hotel near the university always fils up on the evening before football games. History has shown that when the hotel is fully booked, the number of last-minute cancellations has a mean of 6 and standard deviation of 3. The hotel has overbooked FOUR (4) rooms. The average room rate is $100. When the hotel is overbooked, policy is to find a room in a near-by hotel and to pay for the room for the customer. This usually costs the hotel certain amount of money since rooms booked on such late notice are expensive. What would be the rate that the hotel will have to pay to a near-by hotel (in other words, what is the cost of overbooking a room)? (25 points)

Explanation / Answer

Mean, mu = 6

Standard deviation, sigma = 3.

Overbooked rooms, X = 4

Cost of overbooking, Co =?

Cost of under booking, Cu = 100

Standard deviation = 3

Using formula Z = (X - mean) / Standard deviation, sigma

Z = (6 - 4) / 3 = 0.66

Probability corresponding to Z = 0.66 is 0.254 (from standard normal table). This means critical factor = 0.254

We know that

Critical factor = Cu / (Cu + Co)

Implies, 100 / (100+Co) = 0.254

100 = 25.4 + 0.254 Co

Co = (100 – 25.4) / 0.254 = $293.70

Thus, cost of overbooking = $293.70

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