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Chapter 2: Problem 5

For the following numerical variables, state whether each is discrete or continuous. a. The length of a 1-year-old rattlesnake b. The altitude of a location in California selected randomly by throwing a dart at a map of the state c. The distance from the left edge at which a 12 -inch plastic ruler snaps when bent far enough to break d. The price per gallon paid by the next customer to buy gas at a particular station

Short Answer

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

Step by step solution

01

a. Rattlesnake length

The length of a 1-year-old rattlesnake can take any value within a certain range, depending on various factors like genetics, living conditions, and diet. This variable is continuous since it can take any value within that range.
02

b. Altitude of a location in California

The altitude of a random location in California can take any value within the range of the lowest and highest points in the state. This variable is continuous, as the altitude can take any value in that range, not just specific discrete values.
03

c. Distance from the left edge of a snapped ruler

The distance from the left edge where a 12-inch plastic ruler snaps can take any value from 0 to 12 inches, depending on how the ruler breaks. This variable is continuous, as there are no restrictions on which specific values it can take within that range.
04

d. Price per gallon for the next gas customer

The price per gallon paid by the next customer at a gas station typically has a fixed value, set by the gas station. However, the price may include fractions of a cent (e.g., $3.599 per gallon), making it difficult to assign discrete values to this variable. In reality, these values are often rounded when calculating the total amount to be paid, but theoretically, the price per gallon can be considered a continuous variable, as it can take any value within a certain range depending on market factors and pricing decisions.

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Key Concepts

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

Discrete Variables
Discrete variables are an important concept in statistics and are characterized by counting numbers that can only take specific, distinct values. These variables arise in scenarios where measurements can result in countable values, typically entire numbers like 0, 1, 2, and so on. Examples include the number of children in a family, the number of cars in a garage, or the number of books on a shelf.
  • Discrete variables can't take on values with fractions or decimals.
  • They are often the result of counting something.
  • Each value represents a separate category or count.
It's essential to distinguish discrete variables from continuous ones because it impacts how you handle data analysis, including the types of graphs and statistical tests you might use. Understanding whether a variable is discrete or not can help in modeling and hypothesis testing.
Continuous Variables
Continuous variables are variables that can assume an infinite number of values within a given range. Unlike discrete variables, continuous variables can have fractions and decimal values. This makes them vital for measuring something that doesn't naturally round into whole numbers. Consider attributes like height, time, weight, and temperature, where precise measurements are crucial.
  • They can take any value between two points on a numerical scale.
  • They can be very precise, measured in fine increments.
  • These values are subject to small, often non-integer changes.
Understanding continuous variables is critical because it alters how you might analyze data. For example, you can apply different statistical methods such as regression analysis, which assumes continuity in data. Recognizing continuous variables helps in making nearly exact predictions as they allow interpolation.
Numerical Variables
Numerical variables are foundational in statistics and can broadly be classified into two categories: discrete and continuous. They represent data that can be quantified and can perform arithmetic operations. The key distinction between numerical variables is whether or not the data can be incrementally measured or counted.
  • Numerical variables that can be counted without fractions are discrete.
  • Variables that involve measurement and can involve fractions or decimals are continuous.
  • Choosing the right analysis method depends on recognizing the type of numerical variable.
Comprehending the nature of a numerical variable is central to any statistical analysis. It informs the choice of statistical measures, graphical representations, and potential predictive tools. By properly identifying numerical variables, one can draw meaningful insights from the data, optimizing both analysis and results.

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Most popular questions from this chapter

Classify each of the following variables as either categorical or numerical. a. Color of an M\&M candy selected at random from a bag of M\&M's b. Number of green M\&M's in a bag of M\&M's c. Weight (in grams) of a bag of M\&M's d. Gender of the next person to purchase a bag of M\&M's at a particular grocery store

Using the five class intervals 100 to \(<120,120\) to \(<140, \ldots, 180\) to \(<200,\) construct a frequency distribution based on 70 observations whose histogram could be described as follows: a. symmetric b. bimodal c. positively skewed d. negatively skewed

An article in the San Luis Obispo New Times (February 4,2016 ) reported the accompanying concussion $$\begin{array}{|lc|}\hline \text { Sport } & \begin{array}{c}\text { Concussion Rate (Concussions } \\ \text { per } 10,000 \text { athletes) }\end{array} \\\\\hline \text { Football } & 11.2 \\\\\text { Lacrosse (Boys) } & 6.9 \\\\\text { Lacrosse (Girls) } & 5.2 \\\\\text { Wrestling } & 6.2 \\\\\text { Basketball (Girls) } & 5.6 \\\\\text { Basketball (Boys) } & 2.8 \\\\\text { Soccer (Girls) } & 6.7 \\\\\text { Soccer (Boys) } & 4.2 \\\\\text { Field Hockey } & 4.2 \\\\\text { Volleyball } & 2.4 \\\\\text { Softball } & 1.6 \\\\\text { Baseball } & 1.2 \\\\\hline\end{array}$$ a. Construct a dotplot for the concussion rate data. b. In addition to the three girls' sports indicated in the table (lacrosse, basketball, and soccer), the article also reported concussion rates for field hockey, volleyball, and softball, which are girls' sports. Locate the points on the dotplot that correspond to concussion rates for girls" sports and highlight them in a different color. Based on the dotplot, would you say that the concussion rates tend to be lower for girls' sports? (Hint: See Example 2.5.)

For each of the five data sets described, answer the following three questions and then use Figure 2.2 to choose an appropriate graphical display for summarizing the data. Question 1: How many variables are in the data set? Question 2 : Is the data set categorical or numerical? Question 3: Would the purpose of the 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 credit card debt of students at a college, the financial aid office asks each student in a random sample of 75 students about his or her amount of credit card debt. Data Set 2: To learn about how number of hours worked per week and number of hours spent watching television in a typical week are related, each person in a sample of size 40 was asked to keep a log of hours worked and hours spent watching television for one week. At the end of the week, each person reported the total number of hours spent on each activity. Data Set 3: To see if satisfaction level differs for airline passengers based on where they sit on the airplane, all passengers on a particular flight were surveyed at the end of the flight. Passengers were grouped based on whether they sat in an aisle seat, a middle seat, or a window seat. Each passenger was asked to indicate his or her satisfaction with the flight by selecting one of the following choices: very satisfied, satisfied, dissatisfied, and very dissatisfied. Data Set 4: To learn about where students purchase textbooks, each student in a random sample of 200 students at a particular college was asked to select one of the following responses: campus bookstore, off-campus bookstore, purchased all books online, or used a combination of online and bookstore purchases. Data Set 5: To compare the amount of money men and women spent on their most recent haircut, each person in a sample of 20 women and each person in a sample of 20 men was asked how much was spent on his or her most recent haircut.

Gave the following data on saturated fat (in grams), sodium (in \(\mathrm{mg}\) ), and calories for 36 fast-food items. a. Construct a scatterplot using \(y=\) calories and \(x=\) fat. Does it look like there is a relationship between fat and calories? Is the relationship what you expected? Explain. b. Construct a scatterplot using \(y=\) calories and \(x=\) sodium. Write a few sentences commenting on the difference between this scatterplot and the scatterplot from Part (a). c. Construct a scatterplot using \(y=\) sodium and \(x=\) fat. Does there appear to be a relationship between fat and sodium? d. Add a vertical line at \(x=3\) and a horizontal line at \(y=\) 900 to the scatterplot in Part (c). This divides the scatterplot into four regions, with some points falling into each region. Which of the four regions corresponds to healthier fast-food choices? Explain.
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