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Chapter 4: Problem 34

The U.S. Department of Transportation reported that during November, \(83.4 \%\) of Southwest Airlines' flights, \(75.1 \%\) of US Airways' flights, and \(70.1 \%\) of JetBlue's flights arrived on time (USA Today, January 4, 2007). Assume that this on-time performance is applicable for flights arriving at concourse A of the Rochester International Airport, and that \(40 \%\) of the arrivals at concourse A are Southwest Airlines flights, \(35 \%\) are US Airways flights, and \(25 \%\) are JetBlue flights. a. Develop a joint probability table with three rows (airlines) and two columns (on-time arrivals vs. late arrivals). b. \(\quad\) An announcement has just been made that Flight 1424 will be arriving at gate 20 in concourse A. What is the most likely airline for this arrival? c. What is the probability that Flight 1424 will arrive on time? d. Suppose that an announcement is made saying that Flight 1424 will be arriving late. What is the most likely airline for this arrival? What is the least likely airline?

Short Answer

Expert verified
a. Joint probability table created. b. Southwest Airlines is most likely. c. Probability of on-time: 0.7717. d. Most likely late: US Airways; least likely late: Southwest.

Step by step solution

01

Understand the Given Information

We're given three airlines' on-time arrival percentages and their respective proportions of incoming flights to concourse A at Rochester International Airport. This information will be used to calculate joint probabilities.
02

Create the Joint Probability Table

To develop the joint probability table, calculate the probabilities of on-time and late arrivals for each airline. Multiply the probability of an airline's flights by their on-time and late percentages:- **Southwest Airlines**: - On-time: \(0.40 \times 0.834 = 0.3336\) - Late: \(0.40 \times 0.166 = 0.0664\)- **US Airways**: - On-time: \(0.35 \times 0.751 = 0.26285\) - Late: \(0.35 \times 0.249 = 0.08715\)- **JetBlue**: - On-time: \(0.25 \times 0.701 = 0.17525\) - Late: \(0.25 \times 0.299 = 0.07475\)Joint probability table:| Airline | On-time | Late ||---------------|---------|---------|| Southwest | 0.3336 | 0.0664 || US Airways | 0.26285 | 0.08715 || JetBlue | 0.17525 | 0.07475 |
03

Determine the Most Likely Airline for a Random Flight

The most likely airline for a random flight is the one with the highest overall proportion of arrivals. Check the highest value in the table considering both 'On-time' and 'Late' columns.- Southwest: \(0.3336 + 0.0664 = 0.40\)- US Airways: \(0.26285 + 0.08715 = 0.35\)- JetBlue: \(0.17525 + 0.07475 = 0.25\)Southwest Airlines is the most likely airline.
04

Calculate the Probability of On-time Arrival for Flight 1424

Sum the probabilities of on-time arrivals for all airlines:\(0.3336 + 0.26285 + 0.17525 = 0.7717\)Therefore, the probability that Flight 1424 will arrive on time is 0.7717.
05

Analyze Late Arrivals to Determine Most and Least Likely Airlines

For late flights, identify the airline with the highest and lowest probability of being late: - Southwest: 0.0664 - US Airways: 0.08715 - JetBlue: 0.07475 - Most likely late: US Airways - Least likely late: Southwest Airlines

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Key Concepts

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

Joint Probability Table
A joint probability table is a useful tool in statistics. It helps us visualize and calculate the likelihood of different outcomes for two or more events. In this exercise, we are dealing with flights from three airlines: Southwest, US Airways, and JetBlue. The table captures the probability of flights arriving on time or late for each airline.
To build such a table, you need two pieces of information for each airline:
  • The proportion of incoming flights to the airport.
  • The on-time arrival percentage.
To fill out the table, multiply the overall proportion of an airline's flights by their respective on-time and late percentages. For instance, with Southwest, which has 83.4% on-time arrivals and makes up 40% of arrivals, we calculate:
For on-time: \(0.40 \times 0.834 = 0.3336\)
For late: \(0.40 \times 0.166 = 0.0664\).
Repeat this for each airline to fill out the table. The values in the table reflect the joint probabilities, showing the likelihood of a particular airline having a flight that is on time or late.
On-time Arrivals
Understanding on-time arrival statistics is crucial for analyzing airline performance. In the context of this problem, on-time arrival percentages give us insight into how reliably a flight from a particular airline reaches its destination at the scheduled time.
For Southwest Airlines, which has an on-time percentage of 83.4%, this means that out of every 100 flights, about 83 will arrive according to schedule. For US Airways, with a 75.1% on-time rate, about 75 flights out of every 100 will be punctual. Similarly, JetBlue鈥檚 rate is approximately 70.1%. These statistics paint a picture of how each airline fares in terms of efficiency regarding on-time performance.
Analyzing on-time arrivals involves looking at historical data. Airlines typically strive to improve these metrics by optimizing flight operations and managing disruptions efficiently.
Airline Statistics
Understanding airline statistics, such as the percentages provided for arrivals, is important for passengers and industry analysts alike. These statistics provide insight into the overall performance of airlines and their ability to maintain schedules, which is particularly important for time-sensitive travelers.
In this scenario, we are told that Southwest Airlines, US Airways, and JetBlue have different on-time rates and varying shares of flights arriving at the specified airport. This data allows us to create a predictive model about flight performance, which customers might use to gauge reliability.
Airline statistics not only reflect punctuality but can also affect an airline's reputation and customer satisfaction. Thus, airlines with higher on-time percentages are often favored by those looking for reliability.
Probability Calculation
Probability calculation is the heart of predicting outcomes with uncertain information. In this exercise, different probabilities are calculated to understand the likelihood of specific events. For example, the probability of Flight 1424 arriving on time is calculated by summing the on-time probabilities for all airlines: \(0.3336 + 0.26285 + 0.17525 = 0.7717\), indicating a 77.17% chance of on-time arrival.
Furthermore, to determine the most or least likely airline for a random flight, or for a late-arriving flight, we sum across the corresponding columns to see which airline has the higher or lesser probabilities.
Using these calculations helps us make informed decisions based on statistical data. This way, we gain insight into potential future events, informed by past performance and current conditions.

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Most popular questions from this chapter

In the city of Milford, applications for zoning changes go through a two-step process: a review by the planning commission and a final decision by the city council. At step 1 the planning commission reviews the zoning change request and makes a positive or negative recommendation concerning the change. At step 2 the city council reviews the planning commission's recommendation and then votes to approve or to disapprove the zoning change. Suppose the developer of an apartment complex submits an application for a zoning change. Consider the application process as an experiment. a. How many sample points are there for this experiment? List the sample points. b. Construct a tree diagram for the experiment.

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. \\[ \begin{aligned} P(\text { high-quality oil }) &=.50 \\ P(\text { medium-quality oil }) &=.20 \\ P(\text { no oil }) &=.30 \end{aligned} \\] a. What is the probability of finding oil? b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test follow. \\[ \begin{aligned} P(\text { soil } | \text { high-quality oil }) &=.20 \\ P(\text { soil } | \text { medium-quality oil }) &=.80 \\ P(\text { soil } | \text { no oil }) &=.20 \end{aligned} \\] How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?

Two Wharton professors analyzed 1,613,234 putts by golfers on the Professional Golfers Association (PGA) Tour and found that 983,764 of the putts were made and 629,470 of the putts were missed. Further analysis showed that for putts that were made, \(64.0 \%\) of the time the player was attempting to make a par putt and \(18.8 \%\) of the time the player was attempting to make a birdie putt. And, for putts that were missed, \(20.3 \%\) of the time the player was attempting to make a par putt and \(73.4 \%\) of the time the player was attempting to make a birdie putt (Is Tiger Woods Loss Averse? Persistent Bias in the Face of Experience, Competition, and High Stakes, D.G. Pope and M. E. Schweitzer, June 2009, The Wharton School, University of Pennsylvania). a. What is the probability that a PGA Tour player makes a putt? b. Suppose that a PGA Tour player has a putt for par. What is the probability that the player will make the putt? c. Suppose that a PGA Tour player has a putt for birdie. What is the probability that the player will make the putt? d. Comment on the differences in the probabilities computed in parts (b) and (c).

Assume that we have two events, \(A\) and \(B\), that are mutually exclusive. Assume further that we know \(P(A)=.30\) and \(P(B)=.40\). a. What is \(P(A \cap B) ?\) b. What is \(P(A | B) ?\) c. \(\quad\) A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer. d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

Statistics from the 2009 Major League Baseball season show that there were 157 players who had at least 500 plate appearances. For this group, 42 players had a batting average of 300 or higher, 53 players hit 25 or more home runs, and 14 players had a batting average of .300 or higher and hit 25 or more home runs. Only four players had 200 or more hits (ESPN website, January 10,2010 ). Use the 157 players who had at least 500 plate appearances to answer the following questions. a. What is the probability that a randomly selected player had a batting average of .300 or higher? b. What is the probability that a randomly selected player hit 25 or more home runs? c. Are the events having a batting average of .300 or higher and hitting 25 or more home runs mutually exclusive? d. What is the probability that a randomly selected player had a batting average of .300 or higher or hit 25 or more home runs? e. What is the probability that a randomly selected player had 200 or more hits? Does obtaining 200 or more hits appear to be more difficult than hitting 25 or more home runs? Explain.
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