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The relative frequency table below displays the distribution of annual total personal income (in 2009 inflation-adjusted dollars) for a representative sample of 96,420,486 Americans. These data come from the American Community Survey for 2005-2009. This sample is comprised of \(59 \%\) males and \(41 \%\) females. \({ }^{63}\) (a) Describe the distribution of total personal income. (b) What is the probability that a randomly chosen US resident makes less than $$\$ 50,000$$ per year? (c) What is the probability that a randomly chosen US resident makes less than $$\$ 50,000$$ per year and is female? Note any assumptions you make. (d) The same data source indicates that \(71.8 \%\) of females make less than $$\$ 50,000$$ per year. Use this value to determine whether or not the assumption you made in part (c) is valid. $$\begin{array}{lr}\hline \text { Income } & \text { Total } \\ \hline \$ 1 \text { to } \$ 9,999 \text { or loss } & 2.2 \% \\ \$ 10,000 \text { to } \$ 14,999 & 4.7 \% \\ \$ 15,000 \text { to } \$ 24,999 & 15.8 \% \\ \$ 25,000 \text { to } \$ 34,999 & 18.3 \% \\ \$ 35,000 \text { to } \$ 49,999 & 21.2 \% \\ \$ 50,000 \text { to } \$ 64,999 & 13.9 \% \\ \$ 65,000 \text { to } \$ 74,999 & 5.8 \% \\ \$ 75,000 \text { to } \$ 99,999 & 8.4 \% \\ \$ 100,000 \text { or more } & 9.7 \% \\ \hline\end{array}$$

Step by step solution

01

Analyze the Distribution

To describe the distribution of total personal income, we can observe the relative frequencies provided in the table. The majority of individuals earn between $15,000 and $49,999, with these brackets comprising 55.3% of incomes. This suggests that most Americans fall into the lower to middle-income brackets, indicating a possibly right-skewed distribution with fewer individuals earning the highest incomes.

02

Calculate Probability for Income Less Than $50,000

To find the probability that a randomly chosen US resident makes less than \(50,000, sum the percentages of the income brackets below \)50,000: \(2.2\% + 4.7\% + 15.8\% + 18.3\% + 21.2\% = 62.2\%\). Thus, the probability is 0.622.

03

Calculate Probability for Both Less Than $50,000 and Female

Assuming independence between gender and income bracket, we calculate the probability that a resident makes less than \(50,000 and is female by multiplying the probability of being female (41%) with the probability of making less than \)50,000 (62.2%): \(0.41 \times 0.622 = 0.255\). Thus, the joint probability is 0.255.

04

Validate Assumption in Step 3

To determine if our independence assumption is valid, compare the expected probability of females earning less than $50,000 (0.255) to the actual data (71.8% or 0.718). Since these values differ significantly, it indicates that income and gender are not independent, hence our assumption in Step 3 was invalid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Income Distribution

Income distribution refers to how total income is shared among individuals in a population. In the context of the exercise, it is important to note the concentration of income around certain brackets. Most people earn between $15,000 and $49,999, making up 55.3% of individuals. This data showcases a common characteristic in income distributions. It heavily populates lower income brackets and thins out in higher ranges, leading to a distribution that we often describe as "skewed." This skewness is primarily observable because the bulk of the population earns far less than the extremes at the higher end of the spectrum. Understanding income distribution helps in making sense of economic inequalities and can guide policy-making toward fostering economic balance. Recognizing where majority incomes lie informs decisions on taxation, public spending, and social welfare strategies. In scenarios like this, emphasis is laid on households in the lower deep brackets and how they form a substantial percentage of the overall data distribution.

Relative Frequency

Relative frequency is a crucial concept in probability and statistics, as it provides insight into the commonness of specific outcomes within a distribution. In this exercise, relative frequency is exemplified through the percentage of individuals earning a specific income within the total population. The relative frequencies sum up to 100%, giving a clear picture of how income is distributed among the group. To compute the probability of someone earning less than $50,000, we add the relative frequencies of all the income brackets below this threshold: \(2.2\% + 4.7\% + 15.8\% + 18.3\% + 21.2\% = 62.2\%\). This means about 62.2% of the population earns less than this amount, translating to a probability of 0.622.Using relative frequency makes interpreting and visualizing data distributions intuitive by turning frequencies into proportions of the whole set. It is beneficial for understanding the likelihood of specific events or conditions, offering insights into real-world phenomena.

Independence Assumption

The assumption of independence is pivotal in calculating joint probabilities in statistics. In this scenario, it involves assuming that gender and income level are independent when calculating the probability of being female and earning less than \(50,000. To explore this, the probability of a female (41%) is multiplied by the likelihood of earning less than \)50,000 (62.2%): \(0.41 \times 0.622 = 0.255\). This results in an expected joint probability of 25.5%. However, the actual data tells us that 71.8% of females earn less than $50,000, which deviates significantly from our calculated probability. This discrepancy indicates that the independence assumption does not hold, as there's an evident correlation between gender and earning potential. Understanding these assumptions in probability helps avoid incorrect interpretations and fosters a deeper grasp of the relationships within data.

Right-Skewed Distribution

A right-skewed distribution, or positively skewed distribution, is where data points tail off more on the right side. This is characterized by a small number of high-income earners and a larger number of individuals falling into lower income brackets. In the exercise, the distribution shows a clear right-skew with the majority of people earning below $50,000 and a tapering off at higher incomes such as over $100,000. This type of distribution can often reflect real-world income scenarios where wealth is not evenly distributed, and a few individuals hold substantially more than others. Right-skewed distributions are significant in understanding socioeconomic conditions. They guide understanding of inequality and allow for targeted measures in policy-making to tackle large disparities. Visualizing such distributions can aid in recognizing the need for economic interventions where a small population holds disproportionate wealth relative to the majority. Insights into the nature of this skewness can influence economic policy, helping address the gaps between the wealthiest and the majority population.