At The Kansas City Airport, A Group Of Pilots For Skyways And Yellow Jet Airlines Were Asked Whether Their Flights Were Flying East Or West. The Two-way Table Shows Their Answers.Flight Directions$\[ \begin{tabular}{|c|c|c|c|} \hline & East & West
Introduction
In the world of statistics and data analysis, two-way tables are a crucial tool for understanding complex relationships between variables. A two-way table is a table that displays the frequency distribution of two categorical variables. In this article, we will explore a real-life scenario where a group of pilots from Skyways and Yellow Jet airlines were asked about the direction of their flights. We will use this scenario to understand how to analyze a two-way table and extract meaningful insights from the data.
The Two-Way Table
The two-way table below shows the answers of the pilots from Skyways and Yellow Jet airlines about the direction of their flights.
| East | West | Total | |
|---|---|---|---|
| Skyways | 15 | 10 | 25 |
| Yellow Jet | 8 | 12 | 20 |
| Total | 23 | 22 | 45 |
Analyzing the Two-Way Table
To analyze the two-way table, we need to understand the frequency distribution of the two variables: the airline (Skyways or Yellow Jet) and the direction of the flight (East or West). We can start by calculating the marginal frequencies, which are the frequencies of each variable separately.
Marginal Frequencies
The marginal frequencies for the airline variable are:
- Skyways: 25
- Yellow Jet: 20
The marginal frequencies for the direction variable are:
- East: 23
- West: 22
Conditional Probabilities
To understand the relationship between the airline and the direction of the flight, we need to calculate the conditional probabilities. The conditional probability of a variable given another variable is the probability of the first variable occurring given that the second variable has occurred.
For example, the conditional probability of a flight being East given that it is a Skyways flight is:
P(East|Skyways) = (Number of East flights with Skyways) / (Total number of Skyways flights) = 15 / 25 = 0.6
Similarly, the conditional probability of a flight being West given that it is a Yellow Jet flight is:
P(West|Yellow Jet) = (Number of West flights with Yellow Jet) / (Total number of Yellow Jet flights) = 12 / 20 = 0.6
Independence
To determine if the airline and the direction of the flight are independent, we need to check if the conditional probabilities are equal to the marginal probabilities.
For example, the marginal probability of a flight being East is:
P(East) = (Number of East flights) / (Total number of flights) = 23 / 45 = 0.5111
The conditional probability of a flight being East given that it is a Skyways flight is:
P(East|Skyways) = 0.6
Since the conditional probability is not equal to the marginal probability, we can conclude that the airline and the direction of the flight are not independent.
Conclusion
In this article, we analyzed a two-way table that shows the answers of a group of pilots from Skyways and Yellow Jet airlines about the direction of their flights. We calculated the marginal frequencies, conditional probabilities, and checked for independence. The results showed that the airline and the direction of the flight are not independent. This analysis can be useful in various fields such as aviation, marketing, and social sciences.
Future Research Directions
This analysis can be extended in various ways. For example, we can analyze the two-way table for other variables such as the time of day, weather conditions, or passenger demographics. We can also use more advanced statistical techniques such as regression analysis or decision trees to understand the relationships between the variables.
References
- [1] Agresti, A. (2018). Statistics: The Art and Science of Learning from Data. Pearson Education.
- [2] Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis. Prentice Hall.
Appendix
The two-way table used in this analysis is shown below.
| East | West | Total | ||
|---|---|---|---|---|
| Skyways | 15 | 10 | 25 | |
| Yellow Jet | 8 | 12 | 20 | |
| Total | 23 | 22 | 45 |
Q: What is a two-way table?
A: A two-way table is a table that displays the frequency distribution of two categorical variables. It is a useful tool for understanding complex relationships between variables.
Q: How do I create a two-way table?
A: To create a two-way table, you need to have two categorical variables and their corresponding frequencies. You can use a spreadsheet software like Microsoft Excel or Google Sheets to create the table.
Q: What are marginal frequencies?
A: Marginal frequencies are the frequencies of each variable separately. They are calculated by summing up the frequencies of each category for a particular variable.
Q: What are conditional probabilities?
A: Conditional probabilities are the probabilities of a variable occurring given that another variable has occurred. They are calculated by dividing the frequency of a particular category by the total frequency of the variable.
Q: How do I check for independence?
A: To check for independence, you need to compare the conditional probabilities with the marginal probabilities. If the conditional probabilities are equal to the marginal probabilities, then the variables are independent.
Q: What are some common applications of two-way tables?
A: Two-way tables are commonly used in various fields such as:
- Marketing: to analyze the relationship between customer demographics and purchasing behavior
- Social sciences: to study the relationship between variables such as income, education, and employment
- Aviation: to analyze the relationship between flight directions and airline companies
- Business: to analyze the relationship between sales and customer demographics
Q: What are some common mistakes to avoid when working with two-way tables?
A: Some common mistakes to avoid when working with two-way tables include:
- Not checking for independence: failing to check for independence can lead to incorrect conclusions
- Not using the correct statistical tests: using the wrong statistical tests can lead to incorrect conclusions
- Not interpreting the results correctly: failing to interpret the results correctly can lead to incorrect conclusions
Q: What are some advanced statistical techniques that can be used with two-way tables?
A: Some advanced statistical techniques that can be used with two-way tables include:
- Regression analysis: to analyze the relationship between variables and predict outcomes
- Decision trees: to analyze the relationship between variables and make predictions
- Cluster analysis: to group similar observations together based on their characteristics
Q: What are some real-life examples of two-way tables?
A: Some real-life examples of two-way tables include:
- Customer demographics and purchasing behavior: a company wants to analyze the relationship between customer demographics and purchasing behavior to improve sales
- Flight directions and airline companies: an airline company wants to analyze the relationship between flight directions and airline companies to improve flight schedules
- Income and education: a government agency wants to analyze the relationship between income and education to improve social welfare programs
Conclusion
Two-way tables are a powerful tool for understanding complex relationships between variables. By following the steps outlined in this article, you can create and analyze two-way tables to gain insights into various fields. Remember to check for independence, use the correct statistical tests, and interpret the results correctly to avoid common mistakes.