Problem 66 - 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 of \(\$ 50,000\) or more?

Step-by-Step Solution

Answer

The probability that a household has an annual income of \)50,000 or more is 0.55.

Step 1: Determine Total Number of Households

Sum the number of households in each income category to find the total number of households. The total is given by: \[ 22 + 23 + 17 + 12 + 26 \]

Step 2: Sum the Values

Calculate the sum: \[ 22 + 23 + 17 + 12 + 26 = 100 \].This confirms that there are 100 households in total.

Step 3: Determine Number of Households with Income of \(50,000 or More

Add the number of households in the categories \$ 50,000-74,999 \, \$ 75,000-99,999 \, and \$ 100,000 \text { or more }: \[ 17 + 12 + 26 \]

Step 4: Calculate the Sum

Find the total number of households with an income of \$ 50,000 \text { or more }\: \[ 17 + 12 + 26 = 55 \]

Step 5: Calculate the Probability

Determine the probability by dividing the number of households with an income of \$ 50,000 \text { or more }\ by the total number of households: \[ \frac{55}{100} \]

Step 6: Simplify the Probability

Simplify the fraction: \[ \frac{55}{100} = 0.55 \].So, the probability is 0.55.

Key Concepts

Probability

Probability helps us understand the likelihood of an event happening. In this exercise, we are asked to find the probability that a household earns \(50,000 or more annually. To calculate this, we use the ratio of the number of favorable outcomes (households meeting the income criteria) to the total number of possible outcomes (all surveyed households). This is represented as:
\[\text{Probability} = \frac{\text{Number of favorable households}}{\text{Total number of households}}\]
In this case, we found that 55 households earn \)50,000 or more out of 100 total households. The probability is thus 0.55 or 55%.

Income Distribution

Income distribution examines how income is spread among a population. In the given exercise, we categorized incomes into five brackets:

• \(0 - \)24,999
• \(25,000 - \)49,999
• \(50,000 - \)74,999
• \(75,000 - \)99,999
• \(100,000 or more

Understanding how many households fall into each bracket helps us analyze socioeconomic trends and disparities. In this survey, more households are in the highest income bracket (\)100,000 or more), which can provide insights into the economic status and potential disparities within the community.

Survey Data Analysis

Survey data analysis allows us to interpret information collected from a specific group. Here's the process used in the exercise:

• Collecting data from 100 households across different income ranges.
• Summing the number of households in each category to ensure the total matches the survey size (100).
• Identifying the number of households meeting the specific criterion (income of $50,000 or more).
• Calculating the probability by dividing the number of qualifying households by the total household sample.

Analyzing survey data helps us derive meaningful conclusions and make data-driven decisions. In this case, we determine economic trends within a small population.

Fractions

Fractions reflect parts of a whole and are essential in probability calculations. In this exercise, we identified:
\[\text{Fraction of households with income} \geq \$50,000 = \frac{55}{100} = 0.55\]Breaking down fractions tells us how many parts out of a whole meet a specified criterion. Simplifying fractions when possible (like converting 55/100 to 0.55) makes it easier to interpret and understand the data. Fractions provide a simple means to represent data proportions, crucial for statistical analysis and probability outcomes.