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.

Question

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.

Answer 1

To determine the maximum number of tickets an airline should sell, we need to calculate the confidence interval using the Central Limit Theorem and the de Moivre-Laplace 1/2-correction.

Let's break down the problem step by step:

Step 1: Calculate the mean and standard deviation
The mean of the number of ticket holders who do not show up is given as 10% or 0.1. Therefore, the mean of the number of ticket holders who show up is 1 - 0.1 = 0.9.

The standard deviation of the number of ticket holders who do not show up is given by the square root of the mean times (1 - the mean), which is √(0.1 * (1 - 0.1)) = √(0.09) = 0.3.

Step 2: Determine the sample size (n)
The sample size (n) represents the maximum number of tickets an airline should sell. We want to be approximately 99% confident that all ticket holders who show up can board the plane. This means we need to find the z-value corresponding to a 99% confidence level.

Using a standard normal distribution table, the z-value for a 99% confidence level is approximately 2.33.

Step 3: Calculate the margin of error (E)
The margin of error (E) represents the acceptable difference between the observed proportion (p̂) of ticket holders who show up and the true proportion (p) at a certain confidence level. The formula for the margin of error is:

E = z * √(p̂ * (1 - p̂) / n) + 1/2n

We know that p̂ = 0.9 (mean of ticket holders who show up), and z = 2.33 (z-value for 99% confidence level). We want to solve for n, so we need to rearrange the formula:

E - 1/2n = z * √(p̂ * (1 - p̂) / n)

Square both sides of the equation:

(E - 1/2n)^2 = z^2 * p̂ * (1 - p̂) / n

Expand the equation:

E^2 - 2E/n + 1/4n^2 = z^2 * p̂ * (1 - p̂) / n

Multiply both sides by n:

nE^2 - 2E + 1/4n = z^2 * p̂ * (1 - p̂)

Multiplying both sides by 4 to eliminate the fraction:

4nE^2 - 8E + 1/n = 4z^2 * p̂ * (1 - p̂)

Rearrange the equation:

4nE^2 - 8E + 1/n - 4z^2 * p̂ * (1 - p̂) = 0

This is a quadratic equation in terms of n. Let's solve it numerically.

Step 4: Solve the quadratic equation numerically
Using a numerical method, such as the quadratic formula or a calculator, we will find the value of n that satisfies the equation:

4nE^2 - 8E + 1/n - 4z^2 * p̂ * (1 - p̂) = 0

Once you have solved the equation, you will find the value for n, which represents the maximum number of tickets an airline should sell in order to be approximately 99% confident that all ticket holders who show up can board the plane.