Final answer:
The airline must use binomial distributions and appropriate overbooking strategies to maximize revenue and reduce the risks of overbooking. Calculations for the optimal number of overbookings involve probability thresholds, while confidence intervals and hypothesis testing provide insight into airlines' seating and safety policies.
Step-by-step explanation:
The problem at hand involves calculating the appropriate number of overbookings an airline can make to maximize revenue while minimizing the probability of having more passengers than available seats. To solve it, we must look into binomial distributions and overbooking strategies.
Overbooking Calculations:
1. To ensure at least a 0.90 probability of a full flight, the airline needs to calculate the minimum number of reservations to make considering the 0.95 probability that a passenger with a reserved seat actually shows up.
2. To keep the probability of more passengers arriving than the flight can accommodate below 0.10, we have to find the maximum number of reservations to make.
3. The airline could consider different seating reservation policies based on these probabilities, such as using a statistical model to estimate no-shows or applying a strict cancellation policy to reduce the likelihood of no-shows.
Confidence Interval and Hypothesis Testing:
b. The random variable X represents the number of unoccupied seats on a single flight, while \( \overline{X} \) denotes the mean number of unoccupied seats for a sample of 225 flights.
c. A normal distribution or z-distribution can be used for constructing the confidence interval since the sample size is large (>30).
d. To construct a 92 percent confidence interval for the population mean number of unoccupied seats per flight, we'd use the sample mean, sample standard deviation, and the size of the sample to calculate the margin of error and then add and subtract this from the sample mean.