Airtrans Flight 307 Can Accommodate 50 Passengers

Airtrans Flight 307 Can Accommodate 50 Passengers But The Flight Is

Airtran’s flight #307 can accommodate 50 passengers, but the flight is overbooked, as 52 tickets were sold. Each ticketed passenger can arrive late and miss the flight with a probability of 0.02. What is the probability that no passenger arrives late? What is the probability that exactly one passenger arrives late? What is the probability that Airtran has to pay overbooking fees (and reschedule passengers to different flights)?

Using the attached data set, create two graphs - one for systolic and one for diastolic pressure. Each graph should clearly delineate the three groups. I have performed the exploratory data analysis on the systolic and diastolic blood pressures. But when I did the bar graph, my y-axis was off. On the y-axis, Systolic BP should have a specific range of and Diastolic BP should have a range of specific numbers of 82.3 to 83.3.

Paper For Above instruction

Introduction

The problem of overbooking in air travel is a common challenge faced by airlines worldwide. Overbooking occurs when airlines sell more tickets than available seats, anticipating a certain percentage of passengers will not show up. In this context, understanding the probabilities involved when a certain number of passengers actually arrive late or miss flights is crucial for operational decision-making and financial planning. This paper explores the probability calculations related to overbooking at Airtran Flight #307, which has a capacity of 50 passengers but sold 52 tickets. Additionally, it analyzes the creation of accurate and informative visualizations for blood pressure data, focusing on specific y-axis ranges for systolic and diastolic blood pressures.

Probability Analysis of Overbooking

In the scenario where Airtran Flight 307 has a capacity of 50 passengers, but 52 tickets are sold, the likelihood of passengers arriving late impacts the potential for overbooking issues. Each ticketed passenger has a probability of 0.02 to arrive late and miss the flight. To determine the probability that no passenger arrives late, the binomial probability model is applicable, where the number of trials is 52, and the probability of "success" (a passenger arriving late) is 0.02.

The probability \( P(X=0) \) that no passenger arrives late is calculated as:

\[ P(X=0) = \binom{52}{0} (0.02)^0 (0.98)^{52} = (1) \times 1 \times (0.98)^{52} \approx 0.98^{52} \]

Similarly, the probability that exactly one passenger arrives late is:

\[ P(X=1) = \binom{52}{1} (0.02)^1 (0.98)^{51} = 52 \times 0.02 \times 0.98^{51} \]

Finally, the probability that the airline has to pay overbooking fees occurs when more than two passengers arrive late, i.e., at least three late arrivals, because the capacity is limited to 50 passengers. Therefore, this probability is:

\[ P(X \geq 3) = 1 - P(X=0) - P(X=1) - P(X=2) \]

which involves calculating \( P(X=2) \):

\[ P(X=2) = \binom{52}{2} (0.02)^2 (0.98)^{50} \]

Computing these probabilities provides insight into the likelihood of overbooking-related complications.

Graphical Data Representation and Range Selection

In analyzing blood pressure data, the goal is to generate informative bar graphs for systolic and diastolic pressures, with clear delineation of the three groups. The challenge encountered was setting the y-axis ranges to reflect the actual data distribution accurately.

For systolic blood pressure, a specific range should be used, likely based on observed minimum and maximum values within the dataset to enhance interpretability. Similarly, for diastolic blood pressure, the y-axis should be set from 82.3 to 83.3, focusing precisely on the narrow range where the data points lie.

Proper setting of this y-axis range involves examining the data to determine its actual spread and then configuring the graph axes accordingly. This approach ensures that the variations in blood pressure are visualized effectively, allowing for better differentiation between groups.

Creating Accurate and Useful Graphs

When creating bar graphs or any other visualizations, it is vital to manually specify the y-axis limits within the plotting software or programming environment (e.g., R, Python’s Matplotlib, or Excel). This helps to focus on the relevant data range and avoid misinterpretation caused by inappropriate axis scaling.

For the systolic blood pressure, if the data spans a narrow range, define the y-axis from the minimum to maximum relevant values observed. For diastolic blood pressure, explicitly set the limits from 82.3 to 83.3. These specific ranges improve clarity, highlight differences between groups, and prevent the y-axis from being off or misleading.

Conclusion

Managing overbooking in airline operations requires grasping probabilistic concepts to evaluate risks effectively. The calculation of probabilities for late arrivals helps airlines mitigate financial risks associated with overcapacity. Simultaneously, accurate data visualization, through proper axis setting, enhances data interpretation for clinical or research purposes. Implementing these analytical and visualization strategies leads to more effective decision-making in both aviation management and health data analysis.

References

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