Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The following table gives the 2002 age distribution of the U.S. population: $$\begin{array}{lcccccc}\hline \text { Group } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \begin{array}{l}\text { Age } \\\\\text { (in years) } \end{array} & \text { Under } 5 & 5-19 & 20-24 & 25-44 & 45-64 & 65 \text { and over } \\\\\hline \begin{array}{l} \text { Number (in } \\\\\text { thousands) }\end{array} & 19,527 & 59,716 & 18,611 & 83,009 & 66,088 & 33,590 \\ \hline\end{array}$$ Let the random variable \(X\) denote a randomly chosen age group within the population. Find the probability distribution associated with these data.

Step-by-Step Solution

Answer

The probability distribution of the random variable X for each age group is: P(X=1) ≈ 0.0696 P(X=2) ≈ 0.2131 P(X=3) ≈ 0.0664 P(X=4) ≈ 0.2960 P(X=5) ≈ 0.2355 P(X=6) ≈ 0.1197

Step 1: Calculate the total population

To find the probability distribution, we need to find the total population. We can do this by summing the number of people in each age group. Total population = 19,527 + 59,716 + 18,611 + 83,009 + 66,088 + 33,590

Step 2: Calculate the probability for each age group

Using the total population calculated in step 1, we will divide the number of people in each age group by the total population to find the probability of each age group. Let's denote the probabilities for age groups 1 to 6 as P(X=1), P(X=2), P(X=3), P(X=4), P(X=5), and P(X=6) respectively. P(X=1) = \(\frac{19,527}{Total\ population}\) P(X=2) = \(\frac{59,716}{Total\ population}\) P(X=3) = \(\frac{18,611}{Total\ population}\) P(X=4) = \(\frac{83,009}{Total\ population}\) P(X=5) = \(\frac{66,088}{Total\ population}\) P(X=6) = \(\frac{33,590}{Total\ population}\)

Step 3: Calculate the total population and the probabilities

Let's now calculate the total population and the probability for each age group. Total population = 19,527 + 59,716 + 18,611 + 83,009 + 66,088 + 33,590 = 280,541 P(X=1) = \(\frac{19,527}{280,541}\) ≈ 0.0696 P(X=2) = \(\frac{59,716}{280,541}\) ≈ 0.2131 P(X=3) = \(\frac{18,611}{280,541}\) ≈ 0.0664 P(X=4) = \(\frac{83,009}{280,541}\) ≈ 0.2960 P(X=5) = \(\frac{66,088}{280,541}\) ≈ 0.2355 P(X=6) = \(\frac{33,590}{280,541}\) ≈ 0.1197

Step 4: Write out the probability distribution associated with these data

Now that we have calculated the probabilities for each age group, we can write out the probability distribution for the random variable X. The probability distribution of X is: P(X=1) ≈ 0.0696 P(X=2) ≈ 0.2131 P(X=3) ≈ 0.0664 P(X=4) ≈ 0.2960 P(X=5) ≈ 0.2355 P(X=6) ≈ 0.1197

Key Concepts

Applied Mathematics

When we talk about applied mathematics, we are referring to the use of mathematical methods by different fields such as science, engineering, business, computer science, and industry. In essence, applied mathematics is not just about abstract theories; it is about practical problem-solving.

A relevant applied mathematics concept in this exercise is probability. Probability is employed to predict how likely events are to occur. This is crucial in a multitude of areas, from risk assessment in finance to making predictions in weather forecasts or determining outcomes in genetics.

This textbook exercise leverages probability to analyze a real-world scenario, which is the age distribution of the U.S. population. Such analysis can aid in planning for the educational needs of younger age groups, health care facilities for older populations, and economic forecasting.

Age Distribution Analysis

Analyzing age distribution involves understanding the various age categories within a population and their respective proportions. This is a key demographic analysis that can provide insight into the social structure and future trends of a society.

For instance, a high proportion of young people might suggest a need for more educational resources, while a large elderly population may indicate an increased demand for healthcare services. Governments and businesses often use age distribution analysis for planning and decision-making purposes.

By turning age distribution data into probability distributions as seen in the given exercise, we transform raw data into actionable information. Ever-changing demographics shape policy decisions and market strategies, thereby demonstrating the importance of an accurate and comprehensive age distribution analysis.

Random Variable

In statistics, a random variable is a numerical description of the outcomes of a random phenomenon. It can be thought of as a variable whose values depend on the outcomes of a random process.

A random variable is usually denoted by a capital letter, such as 'X' in this exercise, and it can take on multiple possible values, each with its own probability. The collection of all possible values and their associated probabilities is known as the probability distribution of the random variable.

The exercise provided includes a discrete random variable, where specific age groups are associated with various probabilities. Understanding the concept of a random variable is fundamental when dealing with probabilistic models or any form of data that involves randomness. In the context of age distribution, each group represents a potential outcome for the variable 'age group of a randomly chosen individual from the U.S. population'.