Question

The following table shows the distribution of ages of Americans. \({ }^{3}\) Age distribution in reference population $$ \begin{array}{|cc|} \hline \text { Age } & \text { Proportion } \\ \hline 0-19 & 0.27 \\ 20-29 & 0.14 \\ 30-39 & 0.13 \\ 40-49 & 0.14 \\ 50-64 & 0.19 \\ 65+ & 0.13 \\ \hline \end{array} $$ Find the probability that the age of a randomly chosen American (a) is less than 20 . (b) is between 20 and \(49 .\) (c) is greater than \(49 .\) (d) is greater than \(29 .\)

Short answer

(a) 0.27 (b) 0.41 (c) 0.32 (d) 0.59

Step-by-step solution

Identifying the Age Category for Part (a)

For part (a), we are looking for the probability that the age of a randomly chosen American is less than 20 years. This corresponds to the first age category in the given table.

Calculating Probability for Part (a)

The proportion for the age group 0-19 is provided as 0.27. Since this proportion represents the probability of an American being in this age group, the probability is simply the given proportion.

Identifying the Age Categories for Part (b)

For part (b), we need to consider Americans aged between 20 and 49. This range encompasses three age categories in the table: 20-29, 30-39, and 40-49.

Calculating Probability for Part (b)

Calculate the total probability by summing the proportions of the three age categories: 20-29, 30-39, and 40-49.

Identifying the Age Categories for Part (c)

Part (c) asks for the probability that an American is older than 49. The relevant age categories are 50-64 and 65+.

Calculating Probability for Part (c)

Add the proportions for the age categories 50-64 and 65+ to find the probability for part (c).

Identifying the Age Categories for Part (d)

Part (d) requests the probability that an American is older than 29. This includes all age categories except for 0-19 and 20-29.

Calculating Probability for Part (d)

Sum the proportions of the age categories 30-39, 40-49, 50-64, and 65+ to find the probability for part (d).

Key concepts

Age Distribution

Understanding age distribution is crucial when analyzing demographic data. It refers to the way different age groups are proportionally represented within a given population. These distributions provide insights into the structure of the population, potentially influencing policy making, market trends, and social services.

For instance, in the exercise, the age distribution of Americans is divided into six categories ranging from 0-19 to 65 and above. Each category is associated with a proportion, which reflects the percentage of the entire population within that age bracket. An age distribution table can help identify population trends such as the aging of the population or a youth bulge, which have significant societal implications.

When analyzing such data, it is important to consider the context. For example, a high proportion in the 0-19 age group may indicate a growing population, while a larger proportion in older age groups can suggest an aging population that may need more healthcare services. By studying the age distribution, stakeholders can plan resources and strategies accordingly.

Probability Calculation

Probability calculation is a fundamental aspect of statistics that measures the likelihood of a specific event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 denotes certainty.

In the given exercise, probability calculations are used to determine the chances of an American falling into certain age categories. Here's a closer look:

• For part (a): The exercise simply required referencing the proportion for the age group 0-19, which translates directly to the probability of selecting someone under 20, given at 0.27 or 27%.
• For part (b): The probability for ages 20 to 49 is found by adding the proportions of the 20-29, 30-39, and 40-49 age groups.
• For part (c) and part (d): The process is similar, where the probabilities for the categories 'greater than 49' and 'greater than 29' are found by summing the respective age group proportions.

Properly estimating these probabilities allows for informed predictions and decisions based on the age structure, highlighting the importance of accuracy in statistical data.

Statistical Proportions

Statistical proportions represent a part or segment of a population in relation to the whole, often expressed as a percentage or a decimal. They are integral to understanding the composition of datasets in various contexts, including social sciences, healthcare, and business analytics.

In the context of the exercise, proportions are given for different age ranges within the American population. These figures grant insights into the relative size of each age group, which analysts and policymakers use to determine resource allocation, understand potential market sizes for products, or estimate the burden of age-related services on social systems.

Proportions are also the basis for calculating probabilities in a real-world context. The probability of randomly selecting an individual from a given age group is equivalent to the proportion of the population in that age group. Thus, if you know the statistical proportions, you can easily calculate the chance events for statistical purposes.