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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

The ability to recognize and interpret facial expressions is key to successful human interaction. Could this ability be compromised by sleep deprivation? A 2015 study \(^{58}\) took 18 healthy young adult volunteers and exposed them to 70 images of facial expressions, ranging from friendly to threatening. They were each shown images both after a full night of sleep and after sleep deprivation ( 24 hours of being awake), and whether each individual got a full night of sleep or was kept awake first was randomly determined. The study found that people were much worse at recognizing facial expressions after they had been kept awake. (a) What are the explanatory and response variables? (b) Is this an observational study or a randomized experiment? If it is a randomized experiment, is it a randomized comparative experiment or a matched pairs experiment? (c) Can we conclude that missing a night of sleep hinders the ability to recognize facial expressions? Why or why not? (d) In addition, for the people who had slept, the study found a strong positive association between quality of Rapid Eye Movement (REM) sleep and ability to recognize facial expressions. Can we conclude that better quality of REM sleep improves ability to recognize facial expressions? Why or why not?

Three situations are described at the start of this section, on page \(29 .\) In the third bullet, we describe an association between the amount of salt spread on the roads and the number of accidents. Describe a possible confounding variable and explain how it fits the definition of a confounding variable.

Following the steps given, design a randomized comparative experiment to test whether fluoxetine (the active ingredient in Prozac pills) is effective at reducing depression. The participants are 50 people suffering from depression and the response variable is the change on a standard questionnaire measuring level of depression. (a) Describe how randomization will be used in the design. (b) Describe how a placebo will be used. (c) Describe how to make the experiment double-blind.

How Many People Wash Their Hands After Using the Washroom? In Example 1.10 on page \(16,\) we introduce a study by researchers from Harris Interactive who were interested in determining what percent of people wash their hands after using the washroom. They collected data by standing in public restrooms and pretending to comb their hair or put on make-up as they observed patrons' behavior. \({ }^{27}\) Public restrooms were observed at Turner's Field in Atlanta, Penn Station and Grand Central Station in New York, the Museum of Science and Industry and the Shedd Aquarium in Chicago, and the Ferry Terminal Farmers Market in San Francisco. Of the over 6000 people whose behavior was observed, \(85 \%\) washed their hands. Women were more likely to wash their hands: \(93 \%\) of women washed, while only \(77 \%\) of men did. The Museum of Science and Industry in Chicago had the highest hand-washing rate, while men at Turner's Field in Atlanta had the lowest. (a) What are the cases? What are the variables? Classify each variable as quantitative or categorical. (b) In a separate telephone survey of more than 1000 adults, more than \(96 \%\) said they always wash their hands after using a public restroom. Why do you think there is such a discrepancy in the percent from the telephone survey compared to the percent observed?

In Exercises 1.32 to \(1.35,\) describe the sample and describe a reasonable population. A sociologist conducting a survey at a mall interviews 120 people about their cell phone use.
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