Problem 64 - Algebra and Trigonometry

Question

Based on a survey of annual incomes in 100 households. The following table gives the data. $$\begin{array}{l|ccccc}\text { Income } & \$ 0-24,999 & \$ 25,000-49,999 & \$ 50,000-74,999 & \$ 75,000-99,999 & \$ 100,000 \text { or more } \\\\\hline \begin{array}{l}\text { Number } \\ \text { of households }\end{array} & 22 & 23 & 17 & 12 & 26\end{array}$$ What is the probability that a household has an annual income between \(\$ 25,000\) and \(\$ 74,999,\) inclusive?

Step-by-Step Solution

Answer

0.4 or 40%

Step 1: Understand the given data

The data represents the income distribution of 100 households. The table shows income ranges and the number of households in each range.

Step 2: Identify the relevant income ranges

We need to find the probability of a household having an income between \(25,000 and \)74,999, inclusive. Therefore, consider the income ranges \(25,000-\)49,999 and \(50,000-\)74,999.

Step 3: Extract the number of households in the relevant ranges

From the table, households with incomes \(25,000-\)49,999 are 23, and those with incomes \(50,000-\)74,999 are 17.

Step 4: Sum the households in the relevant ranges

Add the number of households in the two relevant ranges: 23 (for \(25,000-\)49,999) + 17 (for \(50,000-\)74,999).

Step 5: Calculate the total number of relevant households

Add the numbers: 23 + 17 = 40. So, there are 40 households with incomes between \(25,000 and \)74,999.

Step 6: Calculate the probability

Divide the number of households in the range by the total number of households: \(\frac{40}{100} = 0.4\).

Step 7: Express the probability

The probability is 0.4 or 40%.

Key Concepts

Income Distribution

Income distribution refers to how income is divided among different groups or individuals in an economy. In our exercise, we look at how 100 households are distributed across different income brackets. This is important because:

• It helps to understand economic inequality.
• It reveals how wealth is spread in a community.
• It aids in creating economic policies.

In the given problem, income distribution allows us to see the number of households falling into various income categories, ranging from less than \(25,000 to more than \)100,000.

Household Survey

A household survey is a method used to collect information from households. These surveys help gather data on a wide range of topics like income, employment, and living conditions.

• They provide insights into the living standards.
• They help in planning and policy-making.
• They can identify areas needing economic support.

In our exercise, the household survey collected annual income data from 100 households. This data was then categorized into different income ranges, making it easier to analyze and understand income distribution.

Probability

Probability measures how likely an event is to occur. It ranges from 0 to 1, where 0 means the event will not occur, and 1 means it will certainly occur.
In our exercise, we want to find the probability that a household falls into a specific income range.

• First, we identify the relevant income ranges.
• Next, we sum the number of households in those ranges.
• Finally, we divide by the total number of households to get the probability.

For the given problem, we sum the households in the \(25,000-\)49,999 and \(50,000-\)74,999 ranges, getting 40 households. Then, divide 40 by 100 to find the probability: \ (0.4) or 40%. This means there's a 40% chance a household will have an income between \(25,000 and \)74,999.