For any given flight, an airline tries to sell as many tickets as possible. Suppose that on average, 10% of ticket holders fail to show up, all independent of one another. Knowing this, an airline will sell more tickets than there are seats available (i.e., overbook the flight) and hope that there is a sufficient number of ticket holders who do not show up to compensate for its overbooking. Using the Central Limit Theorem, determine n , the maximum number of tickets an airline should sell on a flight with 300 seats so that it can be approximately 99% confident that all ticket holders who do show up will be able to board the plane. Use the de Moivre-Laplace 1/2 -correction in your calculations. Hint: You may have to solve numerically a quadratic equation.
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To determine the maximum number of tickets an airline should sell on a flight with 300 seats, we can use the Central Limit Theorem and the de Moivre-Laplace 1/2-correction.
Let's break down the problem:
1. We know that on average, 10% of ticket holders fail to show up. This means that 90% (0.9) of ticket holders do show up.
2. We want to be approximately 99% confident that all ticket holders who do show up will be able to board the plane. This means that the probability of more people showing up than there are seats should be less than or equal to 1% (0.01).
Now, let's use the de Moivre-Laplace 1/2-correction to find the appropriate value of n:
1. Calculate the standard deviation (σ) of the binomial distribution. Since the probability of success is 0.9 and the number of trials is 300, σ = sqrt(n * p * (1-p)). In this case, σ = sqrt(300 * 0.9 * 0.1) = sqrt(27) ≈ 5.196.
2. Apply the de Moivre-Laplace 1/2-correction by subtracting 0.5 from the mean. In this case, the mean (μ) is n * p = n * 0.9.
3. Convert the problem into a z-score by using the formula z = (x - μ) / σ, where x is the number of seats available.
4. Look up the z-score in a standard normal distribution table (or use a calculator) to find the corresponding cumulative probability (P(Z ≤ z)). Since we want to be approximately 99% confident, we are looking for a cumulative probability of 0.99.
5. Subtract the cumulative probability from 0.5 to account for the de Moivre-Laplace 1/2-correction.
6. Solve the equation 0.5 - P(Z ≤ z) = 0.01 numerically to find the value of n. This equation represents the maximum allowable number of tickets that should be sold.
Please note that solving this equation numerically may require the use of a computer or specialized software.
By following these steps, you should be able to determine the maximum number of tickets an airline should sell on a flight with 300 seats to be approximately 99% confident that all ticket holders who show up can board the plane.