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Household income: The following table shows the median income, in thousands of dollars, of American families for 2000 through 2005. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Income (thousands of dollars) } \\ \hline 2000 & 50.73 \\ \hline 2001 & 51.41 \\ \hline 2002 & 51.68 \\ \hline 2003 & 52.68 \\ \hline 2004 & 54.06 \\ \hline 2005 & 56.19 \\ \hline \end{array} $$ a. Plot the data. Does it appear reasonable to model family income using an exponential function? b. Use exponential regression to construct an exponential model for the income data. c. What was the yearly percentage growth rate in median family income during this period? d. From 2000 through 2005, inflation was about 2.4% per year. If median family income beginning at \(50,730 in 2000 had kept pace with inflation, what would be the median family income in 2005? Round your answer to the nearest \)10. e. Consider a family that has the median income of $56,190 in 2005. Use your answer to part d to determine what percentage increase in income would be necessary in order to bring that family’s income in line with inflation over the time period covered in the table.

Step by step solution

01

Plot the Data

To plot the data, place the years on the x-axis and the corresponding median income on the y-axis. The data is: (2000, 50.73), (2001, 51.41), (2002, 51.68), (2003, 52.68), (2004, 54.06), and (2005, 56.19). Connect the points to visualize the trend in income over time.

02

Assess for Exponential Function

Review the plotted data to see if the income trend appears exponential. An exponential trend would show a curve that becomes steeper over time. Here, the income increases at an increasing rate, which suggests an exponential model might fit the data.

03

Conduct Exponential Regression

Perform exponential regression using the formula \( y = a \cdot b^t \), where \( y \) is income, \( a \) is the initial amount, \( b \) is the growth factor, and \( t \) is the time. The regression will calculate \( a \) and \( b \) based on the given data. Using a graphing calculator or software, the model is approximately \( y = 50.23 \cdot (1.019)^t \).

04

Determine Yearly Growth Rate

From the exponential model \( y = 50.23 \cdot (1.019)^t \), the base \( b = 1.019 \) indicates a growth factor of 1.9%. Therefore, the yearly percentage growth rate in median family income is approximately 1.9%.

05

Calculate Inflation-Paced Income for 2005

Using the formula for yearly inflation, calculate the income for 2005 based on a 2.4% increase per year: \( 50.73 \times (1.024)^5 \). After calculating, the steady inflation income for 2005 is approximately \( 56.95 \) (rounded to the nearest ten, \( 57.0 \) thousand).

06

Compute Percentage Increase Needed

To find the percentage increase necessary for the 2005 income to match inflation-adjusted levels, use the formula: \( \, percentage \ increase = \left(\frac{57.0 - 56.19}{56.19}\right) \times 100\% \). Calculating gives a required increase of about 1.44%.

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Key Concepts

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

Understanding Median Household Income

Median household income represents the middle point of income distribution for households. It means that half of all households earn less than this amount, while the other half earns more. It is a critical indicator of the economic health of families and the overall economy.
Median household income is often preferred over average income because it is not skewed by extremely high or low incomes. This makes it a more accurate reflection of the 'typical' household's earnings.
Analyzing changes in median household income over time helps understand economic growth or decline and monitor changes in living standards. In our case, we look at American families' incomes between 2000 and 2005, which show an upward trend, suggesting economic improvement.

Determining Yearly Growth Rate

The yearly growth rate is a measure of how much the median income increases each year. It is derived from examining the rise in income over a set period. This rate is crucial for understanding how quickly or slowly income is growing in a given timeframe.
To calculate it, we use an exponential regression model that gives us a formula of the form \( y = a \cdot b^t \), where \( a \) is the initial income and \( b \) is the growth factor.
For example, in our analysis, the growth factor is \( 1.019 \), indicating an annual increase of roughly 1.9%. This percentage shows how much more, on average, households earn each year compared to the year before, providing insight into economic trends.

The Impact of Inflation Adjustment

Inflation adjustment is a process of modifying monetary figures to reflect changes in purchasing power. It's important because prices of goods and services generally increase over time, reducing the purchasing power of money.
To maintain consistent purchasing power, incomes need to rise at least at the rate of inflation. In our exercise, the average inflation rate is noted as 2.4% per year from 2000 to 2005.
Using inflation adjustment, we forecast what the 2005 income should have been to match 2000 levels in terms of purchasing power. Calculating this gives an inflation-adjusted median income of approximately $57,000 in 2005, implying that while median income did rise, it didn't fully keep up with inflation pressures during this period.

Exponential Function Modeling

Exponential function modeling helps in predicting future values based on past trends. It's particularly suitable when data shows a consistent rate of growth, such as our household income example.
The exponential model is defined mathematically by the equation \( y = a \cdot b^t \), where \( a \) is the starting point or the initial value, and \( b \) is the base, representing the growth factor. This model captures situations where growth accelerates over time, forming a curve rather than a straight line.
Used in conjunction with regression analysis, this model allows us to fit the income data into a curve and project future incomes. It reveals trends like the steady 1.9% yearly increase in median household income, a vital tool for economic planning and analysis.