Problem 9
Question
Let \(P(x)\) be the cumulative distribution function for the household income distribution in the US in 2006 . Values of \(P(x)\) are in the following table: $$ \begin{array}{|l|c|c|c|c|c|c} \hline \text { Income } x \text { (thousand \$) } & 20 & 40 & 60 & 80 & 100 & 150 \\ \hline P(x)(\%) & 21.7 & 45.4 & 63.0 & 75.8 & 84.0 & 94.0 \\ \hline \end{array} $$ (a) What percent of the households made between \(\$ 40,000\) and \(\$ 60,000\) ? More than \(\$ 150,000\) ? (b) Approximately what was the median income? (c) Is the statement "More than one-third of households made between \(\$ 40,000\) and \(\$ 80,000^{\prime \prime}\) true or false?
Step-by-Step Solution
Verified Answer
(a) 17.6%, 6.0%; (b) Between $40,000 and $60,000; (c) False.
1Step 1: Calculate Percent of Households Making Between $40,000 and $60,000
To find the percentage of households that made between \(40,000 and \)60,000, subtract the cumulative percentage at \(40,000 from the cumulative percentage at \)60,000. From the table, these values are 63.0% for \(60,000 and 45.4% for \)40,000. Thus, the calculation is: \ 63.0\% - 45.4\% = 17.6\%.
2Step 2: Calculate Percent of Households Making More Than $150,000
The cumulative percentage at \(150,000 is 94.0%, representing the percentage of households making \)150,000 or less. To find those making more, subtract this value from 100%: \ 100\% - 94.0\% = 6.0\%.
3Step 3: Determine the Median Income
The median income is the income level where 50% of households earn less, and 50% earn more. In this table, $40,000 has a cumulative percentage of 45.4%, and $60,000 has 63.0%. Since 50% falls between these percentages, the median income must be between $40,000 and $60,000.
4Step 4: Verify Statement: More Than One-Third of Households Made Between $40,000 and $80,000
First, calculate the percentage of households making between \(40,000 and \)80,000 by subtracting the cumulative percentage at \(40,000 from that at \)80,000. These values are 75.8% for \(80,000 and 45.4% for \)40,000. \ 75.8\% - 45.4\% = 30.4\%. \ Since one-third of 100% is approximately 33.3%, 30.4% is less than 33.3%. Thus, the statement is false.
Key Concepts
Household Income DistributionMedian IncomePercent of Households
Household Income Distribution
Household income distribution helps us to understand how income varies across different households in a region or a country. It provides insights into how wealth is shared among a population. In a cumulative distribution function (CDF) like the one given, each value of \(P(x)\) tells us the percentage of households earning less than a particular income bracket.
For instance, the function \(P(x)\) is presented as a series of percentages for various income brackets. In the example given, 21.7% of households earn less than \(20,000, and 45.4% earn less than \)40,000. This data helps form a picture of how income is distributed and allows for comparisons between different segments of the population. By analyzing this distribution, policymakers can identify areas of income disparity and consider implementing equity measures.
Understanding this distribution is vital for a variety of applications such as targeting welfare programs or adjusting tax brackets.
For instance, the function \(P(x)\) is presented as a series of percentages for various income brackets. In the example given, 21.7% of households earn less than \(20,000, and 45.4% earn less than \)40,000. This data helps form a picture of how income is distributed and allows for comparisons between different segments of the population. By analyzing this distribution, policymakers can identify areas of income disparity and consider implementing equity measures.
Understanding this distribution is vital for a variety of applications such as targeting welfare programs or adjusting tax brackets.
Median Income
The median income is a measure of central tendency that indicates the middle point of a distribution, where half the households earn less, and half earn more. It is a crucial metric because it represents the point of balance in the income distribution and is often considered a better indicator than the mean in skewed distributions.
From the cumulative distribution table given, to determine the median income, we identify where the cumulative percentage crosses the 50% mark. In the example provided, between $40,000, with 45.4% of households earning less, and $60,000, with 63.0% earning less, the 50% threshold exists. Therefore, the median income lies somewhere between these two figures.
This value tells us about the center of the household income distribution and signals whether income is growing or shrinking over time. A changing median income can indicate shifts in economic prosperity or hardship.
From the cumulative distribution table given, to determine the median income, we identify where the cumulative percentage crosses the 50% mark. In the example provided, between $40,000, with 45.4% of households earning less, and $60,000, with 63.0% earning less, the 50% threshold exists. Therefore, the median income lies somewhere between these two figures.
This value tells us about the center of the household income distribution and signals whether income is growing or shrinking over time. A changing median income can indicate shifts in economic prosperity or hardship.
Percent of Households
The percentage of households earning within a particular income range can be calculated using the cumulative distribution function. This percentage tells us how many households fall into specific income brackets and offers insights into the economic status of a population segment.
For example, to find the percentage of households earning between $40,000 and $60,000, we subtract the cumulative percentage at $40,000 from the cumulative percentage at $60,000. Thus, 63.0% - 45.4% equals 17.6%, meaning that 17.6% of households earn within this income range.
Similarly, to determine the percentage of households earning more than a certain amount, such as $150,000, one can subtract the cumulative percentage at this income level from 100%. In this example, 100% - 94.0% equals 6.0%, indicating that 6% of households earn more than $150,000. These calculations are essential for understanding economic diversity and can guide economic policies and business strategies.
For example, to find the percentage of households earning between $40,000 and $60,000, we subtract the cumulative percentage at $40,000 from the cumulative percentage at $60,000. Thus, 63.0% - 45.4% equals 17.6%, meaning that 17.6% of households earn within this income range.
Similarly, to determine the percentage of households earning more than a certain amount, such as $150,000, one can subtract the cumulative percentage at this income level from 100%. In this example, 100% - 94.0% equals 6.0%, indicating that 6% of households earn more than $150,000. These calculations are essential for understanding economic diversity and can guide economic policies and business strategies.
Other exercises in this chapter
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