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Chapter 1: Problem 2

The U.S. Department of Energy provides fuel economy information for a variety of motor vehicles. A sample of 10 automobiles is shown in Table 1.6 (Fuel Economy website, February 22,2008 ). Data show the size of the automobile (compact, midsize, or large), the number of cylinders in the engine, the city driving miles per gallon, the highway driving miles per gallon, and the recommended fuel (diesel, premium, or regular). a. How many elements are in this data set? b. How many variables are in this data set? c. Which variables are categorical and which variables are quantitative? d. What type of measurement scale is used for each of the variables?

Short Answer

Expert verified
a) 10 elements b) 5 variables c) Categorical: Size, Fuel type; Quantitative: Cylinders, City MPG, Highway MPG d) Nominal: Size, Fuel; Interval: Cylinders; Ratio: MPG values

Step by step solution

01

Determine the Number of Elements

Elements refer to the individual items or entities in a data set. In this case, each automobile in the sample is an element. Since the data set consists of 10 automobiles, there are 10 elements.
02

Identify the Number of Variables

Variables are the different attributes or characteristics of each element within the data set. The problem mentions five variables: 1) the size of the automobile, 2) the number of cylinders, 3) the city driving miles per gallon, 4) the highway driving miles per gallon, and 5) the recommended fuel type. This means there are five variables in total.
03

Classify Variables as Categorical or Quantitative

Categorical variables represent categories or groups, which for this data set are: size of the automobile and recommended fuel type. Quantitative variables are numeric and can be measured, which include: number of cylinders, city driving miles per gallon, and highway driving miles per gallon.
04

Determine Measurement Scales for Each Variable

- The scale of measurement for 'size of the automobile' and 'recommended fuel' is nominal, as they represent categories without a specific order. - For 'number of cylinders,' the scale is interval since this is a discrete numerical value. - For 'city driving miles per gallon' and 'highway driving miles per gallon,' the scale is ratio because they are continuous values with a true zero point, allowing for meaningful comparison of ratios.

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Key Concepts

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

Measurement Scales
Measurement scales are important in data analysis as they determine how data can be collected, compared, and interpreted. There are four primary types of measurement scales: nominal, ordinal, interval, and ratio. In this context:
  • Nominal Scale: This is used for data that can be categorized but not ranked or ordered. Variables such as 'size of the automobile' (compact, midsize, or large) and 'recommended fuel type' (diesel, premium, or regular) fall under this scale because they represent categories without a numeric or ordered significance.
  • Interval Scale: Although not mentioned directly in the provided exercise, an interval scale is used for numeric values where the difference between values is meaningful, but there is no true zero point. An example might be temperature in Celsius.
  • Ratio Scale: This is used for numeric data that has both a meaningful difference and a true zero point, allowing comparisons in terms of "twice as much" for example. Here, 'city driving miles per gallon' and 'highway driving miles per gallon' fall under this category. They allow for meaningful comparisons such as one vehicle having twice the efficiency of another.
Categorical and Quantitative Variables
In statistics, understanding the types of variables is crucial for analysis.
  • Categorical Variables: These variables place data into specific groups or categories. They are not numerical and often involve labeling. In the exercise, both 'size of the automobile' and 'recommended fuel type' are categorical variables as they classify the automobiles without implying any quantitative measure.
  • Quantitative Variables: These are numerical variables that represent measurable quantities. They are further subdivided into discrete and continuous variables. For example, 'number of cylinders' is a discrete quantitative variable as it involves integer values that can be counted. On the other hand, 'city driving miles per gallon' and 'highway driving miles per gallon' are continuous quantitative variables because they can take any value within a certain range, reflecting the precise efficiency of the vehicles.
Descriptive Statistics
Descriptive statistics are a set of brief descriptive coefficients that summarize a given data set, which can be representative of the entire or a sample of a population. The goal is to provide a simple summary of the sample and the measures.
  • Averages: One of the most common forms in descriptive statistics is the average or mean, which gives an idea of the central tendency. For fuel economy, this might mean calculating the average city or highway miles per gallon across all vehicles in the sample.
  • Measures of Spread: Descriptive statistics also involve understanding the variability of the data, commonly using measures like range or standard deviation. Knowing the spread can help recognize how much vehicles differ in terms of efficiency categories.
  • Frequency Distribution: This involves the arrangement of data to show how frequently each possible set of values occurs. For categorical variables like 'size of the automobile' or 'recommended fuel type', frequency distributions can present how many cars fall into each category.
By focusing on these statistics, one can easily grasp the essential characteristics of the data set, forming a basis for more complex analysis.

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Most popular questions from this chapter

The Energy Information Administration of the U.S. Department of Energy provided time series data for the U.S. average price per gallon of conventional regular gasoline between July 2006 and June 2009 (Energy Information Administration website, June 2009 ). Use the Internet to obtain the average price per gallon of conventional regular gasoline since June 2009 a. Extend the graph of the time series shown in Figure 1.1 b. What interpretations can you make about the average price per gallon of conventional regular gasoline since June \(2009 ?\) c. Does the time series continue to show a summer increase in the average price per gallon? Explain.

The Wall Street Journal \((W S J)\) subscriber survey (October 13,2003 ) asked 46 questions about subscriber characteristics and interests. State whether each of the following questions provided categorical or quantitative data and indicate the measurement scale appropriate for each. a. What is your age? b. Are you male or female? c. When did you first start reading the \(W S J ?\) High school, college, early career, midcareer, late career, or retirement? d. How long have you been in your present job or position? e. What type of vehicle are you considering for your next purchase? Nine response categories include sedan, sports car, SUV, minivan, and so on.

A survey of 430 business travelers found 155 used a travel agent to make travel arrangements (USA Today, November 20, 2003). a. Develop a descriptive statistic that can be used to estimate the percentage of all business travelers who use a travel agent to make travel arrangements. b. The survey reported that the most frequent way business travelers make travel arrangements is by using an online travel site. If \(44 \%\) of business travelers surveyed made travel arrangements this way, how many of the 430 business travelers used an online travel site? c. Are the data on how travel arrangements are made categorical or quantitative?

The Hawaii Visitors Bureau collects data on visitors to Hawaii. The following questions were among 16 asked in a questionnaire handed out to passengers during incoming airline flights in June 2003 \(\cdot\) This trip to Hawaii is my: 1 st, 2 nd, 3 rd, 4 th , and so on. \(\cdot\) The primary reason for this trip is: (10 categories including vacation, convention, honeymoon) \(\cdot\) Where I plan to stay: (11 categories including hotel, apartment, relatives, camping) \(\cdot\) Total days in Hawaii a. What is the population being studied? b. Is the use of a questionnaire a good way to reach the population of passengers on incoming airline flights? c. Comment on each of the four questions in terms of whether it will provide categorical or quantitative data.

The Ritz-Carlton Hotel used a customer opinion questionnaire to obtain performance data about its dining and entertainment services (The Ritz-Carlton Hotel, Naples, Florida, February 2006 ). Customers were asked to rate six factors: Welcome, Service, Food, Menu Appeal, Atmosphere, and Overall Experience. Data were recorded for each factor with 1 for Fair, 2 for Average, 3 for Good, and 4 for Excellent. a. The customer responses provided data for six variables. Are the variables categorical or quantitative? b. What measurement scale is used?
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