The Undesirable Air Lines (UAL???) is notorious for its cost-cutting substandard

Question

The Undesirable Air Lines (UAL???) is notorious for its cost-cutting substandard customer service. Consequently, only 85% of the ticket holders will show up for a flight. The Air Lines workhorse, the Boring 878, has 234 seats. UAL, of course, will make extra money by overbooking. However, they have to worry about the FAA fines, and they have to keep a safety margin, say, that there is at least 95% probability that they can accommodate all those who show up for the Boring 878 flight. How many seats can they actually sell in this case? The Undesirable Air Lines (UAL???) is notorious for its cost-cutting substandard customer service. Consequently, only 85% of the ticket holders will show up for a flight. The Air Lines workhorse, the Boring 878, has 234 seats. UAL, of course, will make extra money by overbooking. However, they have to worry about the FAA fines, and they have to keep a safety margin, say, that there is at least 95% probability that they can accommodate all those who show up for the Boring 878 flight. How many seats can they actually sell in this case? The Undesirable Air Lines (UAL???) is notorious for its cost-cutting substandard customer service. Consequently, only 85% of the ticket holders will show up for a flight. The Air Lines workhorse, the Boring 878, has 234 seats. UAL, of course, will make extra money by overbooking. However, they have to worry about the FAA fines, and they have to keep a safety margin, say, that there is at least 95% probability that they can accommodate all those who show up for the Boring 878 flight. How many seats can they actually sell in this case?

Explanation / Answer

Here, there are 234 seats.

Thus, we need the number of tickets such that, the probability of at most 234 tickets is at least 95%.

Using a normal approximation, if n = the number of tickets to sell, p = 0.85,

Mean (u) = n p = 0.85 n

Standard deviation (s) = sqrt(n p (1 - p)) = 0.357071421sqrt(n)

As for a 95% left tailed area, the critical z score is, by table or technology,

zcrit = 1.644853627

As

x = u + zcrit * s

Then

234 = 0.85n + 1.644853627*0.357071421sqrt(n)

234 = 0.85n + 0.587330222*sqrt(n)

0.85n + 0.587330222*sqrt(n) - 234 = 0

Solving for sqrt(n) by quadratic equation,

sqrt(n) = 16.25009779

Thus,

n = 264.0656782

Thus, they could sell 264 seats. [ANSWER]

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