Question

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

a. Continuous variable \n b. Continuous variable \n c. Continuous variable \n d. Continuous variable.

Step-by-step solution

Determine the nature of variable A

The length of a 1-year-old rattlesnake is a measurement, and can take any non-negative value at appropriate level of precision. This makes it a continuous variable.

Determine the nature of variable B

The altitude of a randomly selected location in California can also take any non-negative value at an appropriate level of precision: mountains, valleys, sea level, etc. Hence this is also a continuous variable.

Determine the nature of variable C

The distance from the left edge at which a 12-inch plastic ruler snaps when bent can take any value between 0 and 12 inches, depending on the precise location of the break. We can measure it to any level of precision, which means this is also a continuous variable.

Determine the nature of variable D

The price per gallon paid by the next customer to buy gas at a particular station can, theoretically, take any positive value (zero or negative prices are not realistic in this case). Even though prices are often measured to the nearest cent, they are not discrete, because there are no gaps between potential price values. As a result, this is also a continuous variable.

Key concepts

Understanding Statistics

Statistics is a crucial field of study that deals with collecting, analyzing, interpreting, presenting, and organizing data. It is a key tool used for a variety of purposes, such as decision-making in business, economics, healthcare, and many other areas. In the context of our exercise, statistics allows us to categorize numerical variables into two types—continuous and discrete. This classification helps with choosing appropriate methods for data analysis since different types of data require different statistical techniques.

For instance, in the exercise, the length of a rattlesnake and the altitude of a location in California are measured continuously, and hence, they are analyzed with statistics suitable for continuous data, such as calculating means, standard deviations, or using regression analysis. Statistics not only aids in understanding the characteristics of these variables but also informs how to meaningfully summarize and use this data in real-life scenarios.

Types of Variables

In statistics, the concept of variables is fundamental. Variables come in various types, each with its own properties and implications for how they should be handled statistically.

Continuous variables are those that can take on an infinite number of values within a given range. They are usually measurements, such as the ones mentioned in our exercise—like the length of a rattlesnake or the altitude of a specific location. These measurements are not restricted to discrete 'steps' but can vary gradually.

On the other hand, discrete variables have distinct, separate values, often represented by counts or integer values. For example, the number of students in a classroom or the number of cars in a parking lot are discrete; one cannot have 2.5 students or 17.3 cars in these contexts. It is important for students to understand the difference since it dictates how one can mathematically manipulate and graph these variables.

Measurement Precision

Precision in measurement refers to the exactness of the values obtained and it deeply influences whether a variable is considered discrete or continuous. Precision is generally dictated by the measuring instrument and the unit of measurement used. The more refined and precise the measuring tool, the more likely the variable is to be continuous.

For example, in the exercise, the distance where the plastic ruler snaps can be measured as finely as the instrument allows—down to fractions of an inch, whereas the price per gallon of gas could theoretically differ by fractions of a cent, which is practically impossible to pay, but possible to calculate. This level of precision affirms their classification as continuous variables. However, even with continuous variables, it is crucial to consider the appropriate level of precision for the context of the measurement, as unnecessary precision would neither add value nor meaning to the data.