Question

Are the variables in Exercises \(10-18\) discrete or continuous? Number of claims filed with an insurance company during a single day.

Short answer

Explain your reasoning. Answer: The number of claims filed with an insurance company during a single day is a discrete variable. This is because claims can only take whole-number values and are countable, as we cannot have a fraction or a decimal for the number of claims filed.

Step-by-step solution

Define the variable

The variable we are working with is the number of claims filed with an insurance company during a single day.

Identify the characteristics of the variable

As this variable represents the number of claims, it can only take whole-number values (e.g., 0, 1, 2, etc.). We cannot have a fraction or a decimal for the number of claims filed.

Categorize the variable

Since the number of claims filed during a single day can only take whole-number values and is countable, we can classify the variable as a discrete variable.

Key concepts

Probability and Statistics

When we dive into the realm of probability and statistics, we're essentially exploring the universe of uncertainty and making educated guesses. Probability allows us to measure the likelihood of an event happening, whereas statistics helps us interpret data and trends from the real world. A foundational concept in both fields is the distinction between different types of variables - specifically, discrete and continuous variables.

Discrete variables are those that can count a finite number of outcomes, such as the flip of a coin (heads or tails), the roll of a die (1 through 6), or the number of textbooks sold at a bookstore each day (0, 1, 2, ...). On the other hand, continuous variables are those that can take on an infinite number of values within a given range, like the temperature in a room or the height of students in a class. Both variables play a pivotal role in data analysis and study designs across various disciplines.

Number of Claims

Understanding the number of claims an insurance company receives over a certain period is critical for assessing risk and predicting future resource needs. The number of claims can vary day-to-day, but even if it fluctuates, each day’s tally is a discrete count. For instance, on Monday, there might be 5 claims, and on Tuesday, there could be 7. It's vital for both insurance professionals and statisticians to analyze these numbers to identify patterns, such as peak claim periods, and to prepare for high-volume days.

By improving data collection practices and employing predictive analytics, insurance companies can optimize their operations and improve customer satisfaction. This predictive power hinges on a sound understanding of whether the variable in question is discrete or continuous, as this affects the choice of statistical methods used for analysis.

Whole-Number Values

The term 'whole-number values' refers to the set of non-negative integers including zero (0, 1, 2, 3, ...). In the context of probability and statistics, many real-world scenarios deal with whole-number values. For example, you can’t have half a car accident or 2.5 tickets sold for a concert. These are instances where the variable we're measuring, like the number of accidents or tickets, is inherently a whole-number.

Specifically, variables that take on whole-number values are, by definition, discrete. They are countable and separate, with no possibility of values between the whole numbers. Recognizing a variable as discrete guides analysts in choosing appropriate mathematical tools, like the Poisson or binomial distributions, which are well-suited for handling countable occurrences in a given interval.