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Income distribution refers to how income is shared among families in a specific area, like Los Angeles in this case. Generally, it is important to understand this distribution to get insights into economic inequality and the standard of living. The key variables to consider are the mean and standard deviation of the income data.

• The mean represents the average income, showing the central value of the distribution.
• The standard deviation indicates how much variation or spread there is from the mean income. A small standard deviation means incomes are close to the mean, while a larger one indicates more variability.

When analyzing income distributions, data is often assumed to follow a normal distribution, also known as a bell curve. Most incomes cluster around the average, with fewer families earning significantly more or less. This assumption helps in simplifying the probability calculations, although it is crucial to verify with real data through skewness and kurtosis measurements.

Probability calculation is the process of determining the likelihood of a given event, such as a family earning more than a certain amount. In our example, we want to calculate the probability that a family earns more than $175,000 annually. The steps are as follows:

• Identify the data range: Find the proportion of families earning up to $175,000.
• Subtract from 1: Since we want families earning more than this amount, we subtract the proportion of those earning less than or equal to the amount from 1. This complement method is common in statistics for calculating probabilities of greater-than situations.

This simple probability calculation gives us insights into income levels in a region and helps make informed economic decisions.

A Z-score, also known as a standard score, is a critical tool for probability calculations in normal distributions. It tells us how far away a specific point is from the mean in terms of standard deviations. For our income scenario, we calculate the Z-score to understand how significantly \(175,000 deviates from the average income.Here’s how it works:

• Formula: The Z-score is calculated as \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the income value of interest (\)175,000), \( \mu \) is the mean income, and \( \sigma \) is the standard deviation.
• Interpretation: The Z-score allows us to place income values on the standard normal distribution. A positive Z-score means the value is above the mean, while a negative score indicates it is below the mean.
• Using Z-tables: This score maps onto a Z-table, which tells us the cumulative probability of incomes up to that value.

Using the Z-score and tables, one can determine the probability of a family earning more than the specified income, thereby understanding income trends and distributions more deeply.