91Ó°ÊÓ

Phoenix is a hub for a large airline. Suppose that on a particular day, 8000 passengers arrived in Phoenix on this airline. Phoenix was the final destination for 1800 of these passengers. The others were all connecting to flights to other cities. On this particular day, several inbound flights were late, and 480 passengers missed their connecting flight. Of these 480 passengers, 75 were delayed overnight and had to spend the night in Phoenix. Consider the chance experiment of choosing a passenger at random from these 8000 passengers. Calculate the following probabilities: a. the probability that the selected passenger had Phoenix as a final destination. b. the probability that the selected passenger did not have Phoenix as a final destination. c. the probability that the selected passenger was connecting and missed the connecting flight. d. the probability that the selected passenger was a connecting passenger and did not miss the connecting flight. e. the probability that the selected passenger either had Phoenix as a final destination or was delayed overnight in Phoenix. f. An independent customer satisfaction survey is planned. Fifty passengers selected at random from the 8000 passengers who arrived in Phoenix on the day described above will be contacted for the survey. The airline knows that the survey results will not be favorable if too many people who were delayed overnight are included. Write a few sentences explaining whether or not you think the airline should be worried, using relevant probabilities to support your answer.

Step by step solution

01

Write the given data into more readable form

- Total Passengers: 8000 - Passengers with Phoenix as a final destination: 1800 - Passengers connecting to other cities: 6200 (8000 - 1800) - Passengers missed connecting flight: 480 - Passengers delayed overnight: 75

02

Calculate Probability for part a

To calculate the probability that the selected passenger had Phoenix as a final destination, we need to divide the number of passengers with Phoenix as their final destination by the total number of passengers. \(P(A) = \frac{1800}{8000}\)

03

Calculate Probability for part b

To calculate the probability that the selected passenger did not have Phoenix as a final destination, we need to divide the number of passengers connecting to other cities by the total number of passengers. \(P(B) = \frac{6200}{8000}\)

04

Calculate Probability for part c

To calculate the probability that the selected passenger was connecting and missed the connecting flight, we need to divide the number of passengers who missed their connecting flight by the total number of passengers. \(P(C) = \frac{480}{8000}\)

05

Calculate Probability for part d

To calculate the probability that the selected passenger was a connecting passenger and did not miss the connecting flight, we first need to find the number of passengers who were connecting and did not miss the connecting flight: it is equal to the total connecting passengers minus passengers who missed their flights, which is \(6200 - 480 = 5720\). Now, divide this number by the total number of passengers: \(P(D) = \frac{5720}{8000}\)

06

Calculate Probability for part e

To calculate the probability that the selected passenger either had Phoenix as a final destination or was delayed overnight in Phoenix, we first need to find the number of passengers who fit in either of these categories: this is equal to the passengers with Phoenix as a final destination plus passengers who were delayed overnight, which is \(1800 + 75 = 1875\). Now, divide this number by the total number of passengers: \(P(E) = \frac{1875}{8000}\)

07

Answer part f using calculated probabilities

Now we'll analyze whether the airline should be worried about the survey based on the calculated probabilities. Among the 50 passengers selected for the survey, we need to analyze how many of them might be delayed overnight. The probability of a randomly selected passenger being delayed overnight is: \(P(F) = \frac{75}{8000}\) For 50 passengers, the expected number of passengers delayed overnight is \(E = 50 \times P(F)\). If this expected number is low, the airline should not worry too much about the survey results, as most passengers had a satisfactory experience. However, if the expected number of delayed overnight passengers is high, the airline should be concerned about the survey results reflecting the negative experiences of these passengers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Analysis

Statistical analysis is an essential tool used to interpret data collected from observations or experiments. It helps us understand trends, make predictions, and inform decision-making. In the context of the Phoenix airline scenario, statistical analysis involves analyzing the data on passenger numbers and their eventual destinations. Knowing the total number of passengers and other metrics, such as missed flights, enables us to compute various probabilities. These probabilities are key to understanding different passenger experiences and planning strategies to improve future operations. By calculating different probabilities, the airline can assess specific aspects like the efficiency of its connections and the satisfaction potential of its services.

Statistics often uses tools like measures of central tendency and variation, regression, and probability. The probabilities calculated in the Phoenix airline situation are derived by simple ratios of favorable outcomes to the total number of passengers. For example, calculating the probability of a passenger having Phoenix as their final destination involves dividing the number of those passengers by the total passengers. This step-by-step method allows for identifying patterns, predicting future occurrences, and making informed adjustments to services.

Passenger Data Analysis

Analyzing passenger data means interpreting and extracting useful insights from the data about passengers and their movements. The data from the Phoenix scenario shows various categories such as passengers with final destinations in Phoenix, those connecting onwards, and those who experienced delays. By categorizing and quantifying these figures, the airline can better understand passenger behavior and operational efficiency.

Passenger data analysis involves sorting passengers into groups based on specific criteria. For instance, distinguishing between connecting passengers who made their flights from those who missed them can help identify bottlenecks in operation. By understanding these numbers, the airline can adopt measures to reduce such occurrences and enhance customer satisfaction. The insights gathered from data analysis not only help in real-time decisions but also in long-term strategic planning.

Probability Calculation

Probability calculations are essential in making statistical predictions and evaluating risks. In passenger analysis, these calculations tell us the likelihood of certain events occurring, such as missing a flight or being delayed. Probabilities can be calculated using simple ratios.

For example, in the Phoenix airline exercise, we calculate different probabilities by considering the ratio of specific outcomes to the total number of passengers. If we want to find the probability of a passenger needing an overnight stay due to a missed flight, we divide the number of such passengers by the total. Thus, each probability gives the airline insight into how common different situations are among its customer base. This information can be instrumental in identifying what measures need to be improved or maintained.

• Basic probability formula:
• \( P(X) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)
• Examples include calculating probabilities for missed connections, final destination arrivals, or delays.

Customer Satisfaction Survey

Surveys are critical in gauging how satisfied customers are with a service or product. In the Phoenix airline exercise, a customer satisfaction survey is planned to measure passengers' experiences, especially of those who were delayed overnight. Conducting such surveys helps airlines understand passengers' satisfaction levels during their travels and hence their reputation.

The airline might be concerned because a high presence of passengers who waited overnight in the survey sample could skew the results negatively. To mitigate this risk, the airline can use probabilities to determine the likelihood of selecting such passengers for the survey. If the probability is high, precautions should be taken in sampling to reflect an actual representative overview of all passengers, not just those with negative experiences. Regular surveys and data analysis can thus inform which areas of the service need improvement and enable airlines to better meet customer needs.