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Chapter 2: Problem 2

For the following numerical variables, state whether each is discrete or continuous. a. The number of insufficient-funds checks received by a grocery store during a given month b. The amount by which a 1 -pound package of ground beef decreases in weight (because of moisture loss) before purchase c. The number of kernels in a bag of microwave popcorn that fail to pop d. The number of students in a class of 35 who have purchased a used copy of the textbook

Short Answer

Expert verified
a. Discrete b. Continuous c. Discrete d. Discrete

Step by step solution

01

a. Insufficient-funds checks

The number of insufficient-funds checks received by a grocery store during a given month can only be whole numbers like 0, 1, 2, etc., as we cannot have a fraction of a check. Therefore, this numerical variable is discrete.
02

b. Weight loss in ground beef

The amount by which a 1-pound package of ground beef decreases in weight due to moisture loss can take on any value within a given range. For example, the weight loss could be 0.2 pounds or 0.357 pounds. Since the weight loss can have infinitely many possible values within a range, this numerical variable is continuous.
03

c. Unpopped popcorn kernels

The number of kernels in a bag of microwave popcorn that fail to pop can only be whole numbers like 0, 1, 2, etc., as we cannot have a fraction of a kernel. Therefore, this numerical variable is discrete.
04

d. Students purchasing used textbooks

The number of students in a class of 35 who have purchased a used copy of the textbook can only be whole numbers between 0 and 35. We cannot have a fraction of a student. Therefore, this numerical variable is discrete.

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Key Concepts

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

Numerical Variables
Numerical variables are fundamental to understanding data in statistics. They represent quantities that can be measured or counted. These variables come in two broad categories: discrete and continuous.
Discrete numerical variables are countable and take on distinct, separate values. An example is the number of students in a class who bought used textbooks. We can only have whole numbers like 0, 1, or 35, as fractional students are not possible. On the flip side, continuous numerical variables can take on an infinite number of values within a given range.
Think of these as variables that can be measured to any level of precision, such as weight or temperature. For instance, the weight lost by a packaged beef due to moisture could be 0.3 pounds or 0.347 pounds. These can be any number, showing its continuous nature.
Data Types
Data types are key to handling and interpreting data correctly. They define the kind of data, which dictates what kind of operations you can perform on the data.
Common data types include:**
  • Numerical: As explained earlier, these include discrete and continuous data.
  • Categorical: These are data that can be divided into distinct categories but have no inherent order. An example could be eye color.
  • Ordinal: Like categorical data but with a meaningful order, such as rankings (first, second, third).
Choosing the correct data type is crucial for analysis. For example, treating a continuous variable like bottle weight as discrete would lead to inaccurate results.
Statistics
Statistics is a branch of mathematics that focuses on data collection, analysis, interpretation, and presentation. It helps us make informed decisions based on data.
Key areas in statistics include:**
  • Descriptive Statistics: Describes and summarizes data through measures like mean, median, mode, and standard deviation.
  • Inferential Statistics: Makes predictions or inferences about a population based on a sample of data.
Understanding statistical methods enables us to discern patterns or trends within data. When dealing with numerical variables like in the exercise, statistics helps us decide which type of variable it is and what conclusions can be drawn based on the data type.

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Most popular questions from this chapter

For each of the five data sets described, answer the following three questions and then use Figure 2.2 to select an appropriate graphical display. Question 1: How many variables are in the data set? Question 2: Are the variables in the data set categorical or numerical? Question 3: Would the purpose of a graphical display be to summarize the data distribution, to compare groups, or to investigate the relationship between two numerical variables? Data Set 1: To learn about the reason parents believe their child is heavier than the recommended weight for children of the same age, each person in a sample of parents of overweight children was asked what they thought was the most important contributing factor. Possible responses were lack of exercise, easy access to junk food, unhealthy diet, medical condition, and other. Data Set 2: To compare commute distances for full-time and part-time students at a large college, commute distance (in miles) was determined for each student in a random sample of 50 full-time students and for each student in a random sample of 50 part-time students. Data Set 3: To learn about how number of years of education and income are related, each person in a random sample of 500 residents of a particular city was asked how many years of education he or she had completed and what his or her annual income was. Data Set 4: To see if there is a difference between faculty and students at a particular college with respect to how they commute to campus (drive, walk, bike, and so on), each person in a random sample of 50 faculty members and each person in a random sample of 100 students was asked how he or she usually commutes to campus. Data Set 5: To learn about how much money students at a particular college spend on textbooks, each student in a random sample of 200 students was asked how much he or she spent on textbooks for the current semester.

The report "2013 International Bedroom Poll: Summary of Findings" describes a survey of 251 adult Americans conducted by the National Sleep Foundation (www.sleep foundation.org/sites/default/files/RPT495a.pdf, retrieved April 15,2017 ). Participants in the survey were asked how often they change the sheets on their bed and were asked to respond with one of the following categories: more than once a week, once a week, every other week, every three weeks, or less often than every three weeks. For this group, \(10 \%\) responded more than once a week, \(53 \%\) responded once a week, \(26 \%\) responded every other week, \(5 \%\) responded every three weeks, and \(6 \%\) responded less often than every three weeks. a. Use the given information to make a relative frequency distribution for the responses to the question. b. Summarize the given information by constructing a bar chart. c. The report also summarized data from 250 adults in Japan. For this group, \(11 \%\) responded more than once a week, \(30 \%\) responded once a week, \(22 \%\) responded every other week, \(9 \%\) responded every three weeks, and \(28 \%\) responded less often than every three weeks. Construct a comparative bar chart that will allow you to compare the response distributions for the U.S. sample and the Japan sample. Comment on similarities and differences between the distributions for these two countries.

For each of the five data sets described, answer the following three questions and then use Figure 2.2 to select an appropriate graphical display. Question 1: How many variables are in the data set? Question 2: Are the variables in the data set categorical or numerical? Question 3: Would the purpose of a graphical display be to summarize the data distribution, to compare groups, or to investigate the relationship between two numerical variables? Data Set 1: To learn about the heights of five-year-old children, the height of each child in a sample of 40 five-year-old children was measured. Data Set 2: To see if there is a difference in car color preferences of men and women, each person in a sample of 100 males and each person in a sample of 100 females was shown pictures of a new model car in five different colors and asked to select which color they would choose if they were to purchase the car. Data Set 3: To learn how GPA at the end of the freshman year in college is related to high school GPA, both high school GPA and freshman year GPA were determined for each student in a sample of 100 students who had just completed their freshman year at a particular college. Data Set 4: To learn how the amount of money spent on a fast-food meal might differ for men and women, the amount spent on lunch at a particular fast-food restaurant was determined for each person in a sample of 50 women and each person in a sample of 50 men. Data Set 5 : To learn about political affiliation (Democrat, Republican, Independent, and Other) of students at a particular college, each student in a random sample of 200 students was asked to indicate his or her political affiliation.

Classify each of the following variables as either categorical or numerical. a. Weight (in ounces) of a bag of potato chips b. Number of items purchased by a grocery store customer c. Brand of cola purchased by a convenience store customer d. Amount of gas (in gallons) purchased by a gas station customer e. Type of gas (regular, premium, diesel) purchased by a gas station customer

The report "Trends in Education 2010: Community Colleges" (www.collegeboard.com/trends) included the accompanying information on student debt for students graduating with an AA degree from a public community college in 2008 . $$\begin{array}{|lc|}\hline \text { Debt } & \text { Relative Frequency } \\\\\hline \text { None } & 0.62 \\\\\text { Less than } \$ 10,000 & 0.23 \\ \text { Between } \$ 10,000 \text { and } \$ 20,000 & 0.10 \\\\\text { More than } \$ 20,000 & 0.05 \\\\\hline\end{array}$$ a. Use the given information to construct a bar chart. b. Comment on student debt for public community college graduates.
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