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Chapter 1: Problem 17

Seven of the ten largest cities in the world are in the Eastern Hemisphere (including the largest: Tokyo, Japan) and three are in the Western Hemisphere. \(^{11}\) Table 1.4 shows the populations, in millions of people, for these cities. (a) How many cases are there in this dataset? How many variables are there and what are they? Is each categorical or quantitative? (b) Display the information in Table 1.4 as a dataset with cases as rows and variables as columns. $$ \begin{array}{ll} \hline \text { Eastern hemisphere: } & 37,26,23,22,21,21,21 \\ \text { Western hemisphere: } & 21,20,19 \\ \hline \end{array} $$

Short Answer

Expert verified
There are 10 cases (cities) in the dataset. The variables are Hemisphere (categorical) and Population (quantitative). The data, when reorganized into a dataset with cases as rows and variables as columns, assigns each city to a specific Hemisphere and provides a Population value (in millions) for each case.

Step by step solution

01

Identify Cases and Variables

Looking at the dataset, the number of 'cases' can be identified as each individual city. Given that there are ten cities mentioned (seven from Eastern Hemisphere and three from Western Hemisphere), there are ten cases in this dataset. The variables in the dataset are 'Hemisphere' and 'Population.' The 'Hemisphere' variable is categorical as it groups the cities into two distinct categories: Eastern and Western hemispheres. The 'Population' variable is quantitative as it provides a numerical value depicting the number of people (in millions) who reside within each city.
02

Reformat Dataset

To display the information in the given table 1.4 as a dataset with cases as rows and variables as columns, assign each city to a row and the related variables (Hemisphere and Population) to columns. It should be noted that the exercise does not provide the specific names for all of the cities, and thus they cannot be individually listed. However, the reformation can be generally understood like so: \[ \begin{tabular}{|c|c|c|} \hline \text{City} & \text{Hemisphere} & \text{Population} \\ \hline \text{City 1} & \text{Eastern} & \text{37} \\ \text{City 2} & \text{Eastern} & \text{26} \\ \vdots & \vdots & \vdots \\ \text{City 10} & \text{Western} & \text{19} \\ \hline \end{tabular}\]
03

Summary of Dataset

In this dataset the cases are individual cities and the variables are 'Hemisphere' (categorical) and 'Population' (quantitative). There are 10 cities in total, with 7 in the Eastern Hemisphere and 3 in the Western Hemisphere. The populations of these cities, listed in millions, ranges from 19 to 37.

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Key Concepts

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

Quantitative Data
Quantitative data refers to data that is numerical and can be measured. In statistical analysis, this type of data is critical because it allows for mathematical computations and provides insight into numerical trends and patterns. In the context of our example with the largest cities in the world, the 'Population' variable represents quantitative data. This is because it consists of numerical values expressed in millions of people living in each city.

Quantitative data helps us to:
  • Identify numerical relationships between different data points, such as the population difference between two cities.
  • Conduct statistical tests to infer patterns or predict outcomes.
  • Visualize data trends over time or across categories using graphs and charts.
In the given dataset, each city's population is listed numerically. The data, ranging from 19 million to 37 million, can be used for various calculations, like finding the average population size or determining the city with the highest population.
Categorical Data
Categorical data refers to data that can be divided into distinct categories or groups without any quantitative value. Instead of showing numerical relationships, this data categorizes information into discrete sections. For instance, in our exercise related to city populations, the variable 'Hemisphere' is an example of categorical data. Here, it's used to group the cities based on their geographic hemispheres鈥擡astern or Western.

Characteristics of categorical data include:
  • It describes characteristics such as categories, labels, or names.
  • Cannot be used for arithmetic operations like addition or multiplication.
  • Commonly analyzed using frequencies and percentages.
In our city example, dividing the dataset into Eastern and Western hemispheres provides a way to analyze population distribution based on geography. By understanding which group a city belongs to, we can explore patterns like which hemisphere tends to have larger cities by population.
Data Representation
Data representation involves organizing and structuring data in a way that is comprehensible and accessible for analysis. This concept is essential in statistical analysis as it ensures clarity and facilitates smoother data interpretation. The goal is to take raw data and put it in a format that highlights patterns and insights.

Effective data representation can take the form of:
  • Tables that organize data into rows and columns, allowing for straightforward comparisons.
  • Charts and graphs that visually depict data trends and relationships.
  • Lists or bullet points, which break down information into manageable parts.
In the exercise with city populations, the dataset is initially presented as a simple list of numbers. However, reorganizing it into a table with rows for each city and columns for the variables 'City', 'Hemisphere', and 'Population' provides a clearer picture of how the data points relate to one another. This setup allows for easier identification of patterns, such as population distributions between Eastern and Western hemispheres, aiding in more efficient data analysis.

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Most popular questions from this chapter

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