91Ó°ÊÓ

Education and income: According to the U.S. Census Bureau, in 2004 the median (middle of the range) annual income of a high school graduate with no further education was about \(\$ 37,000\) per year. If we assume that it takes 4 years to earn a bachelor's degree and 2 additional years to earn a master's degree, the median annual income increases by \(14 \%\) for each year spent toward successful completion of a degree. This describes a function \(I=I(y)\), where \(I\) is the median income in 2004 for a person with y years of college education. a. Express in functional notation the median income in 2004 of an individual with an associate degree, which requires 2 years of college. Calculate the value. b. Make a table of values that shows the median income in 2004 for individuals completing 0 through 4 years of college. c. Make a graph of median income versus years of college completed. Be sure to provide appropriate labels. d. Verify that the formula \(I=37 \times 1.14^{y}\) thousand dollars, where \(y\) is the number of years of college completed, gives the same values as those you found in the table you made.e. Using the formula given in part d, find the median income in 2004 of an individual who has a master's degree. f. Assuming it takes 3 years beyond the master's degree to complete a Ph.D., and assuming the formula in part \(d\) applies, what was the median income of a Ph.D. in 2004? g. In fact, the same rate of increase does not apply for the years spent on a Ph.D. The actual median income for a Ph.D. in 2004 was \(\$ 100,000\). Does that mean that the increase in median income for years spent earning a Ph.D. is higher or lower than \(14 \%\) per year?

Step by step solution

01

Understanding the Question

We are given that the base median income for a high school graduate is \(\$37,000\). With each additional year of college, the income increases by \(14\%\). We'll express and calculate the income for different levels of education and validate the provided formula \( I = 37 \times 1.14^y \).

02

Calculate Income for Associate Degree

For an associate degree, complete 2 years of college. Using the formula \( I = 37 \times 1.14^y \), where \( y = 2 \), we find:\[I(2) = 37 \times 1.14^2 \approx 48.1 \text{ thousand dollars}.\]Thus, the median income is approximately \( \$48,100 \).

03

Creating Income Table for 0-4 Years of College

We'll calculate incomes for each year from 0 to 4, using \( I = 37 \times 1.14^y \):- \( y = 0: I(0) = 37 \times 1.14^0 = 37 \, \text{thousand} \)- \( y = 1: I(1) = 37 \times 1.14^1 \approx 42.2 \, \text{thousand} \)- \( y = 2: I(2) \approx 48.1 \, \text{thousand} \)- \( y = 3: I(3) \approx 54.8 \, \text{thousand} \)- \( y = 4: I(4) \approx 62.5 \, \text{thousand} \).

04

Plotting the Graph

Create a graph with 'Years of College Completed' on the x-axis ranging from 0 to 4, and 'Median Income (\$1000s)' on the y-axis. Plot the points: \((0, 37), (1, 42.2), (2, 48.1), (3, 54.8), (4, 62.5)\) and draw a smooth curve through these points.

05

Verifying the Given Formula

We have used the formula \( I = 37 \times 1.14^y \) to determine incomes previously, which match the expected values, thus verifying the accuracy of the formula.

06

Find Income for Master's Degree

A master's degree requires 6 years total (4 years bachelor's + 2 years additional). Using \( y = 6 \) in \( I = 37 \times 1.14^y \), calculate:\[I(6) = 37 \times 1.14^6 \approx 86.9 \, \text{thousand dollars}.\]

07

Calculate Income for Ph.D.

For a Ph.D., add 3 years beyond the master's for a total of 9 years of college. Using \( y = 9 \), find:\[I(9) = 37 \times 1.14^9 \approx 126.5 \, \text{thousand dollars}.\]The real income is \( \$100,000 \), so the formula overestimates this amount.

08

Compare Increase Rate for Ph.D.

The actual income for a Ph.D. was \( \\(100,000\), lower than \( \\)126,500\) predicted. Therefore, the effective rate of increase for these years is lower than \(14\%\) per year.

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Key Concepts

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

Educational Attainment Impact

Educational attainment significantly influences income levels. When comparing individuals with different education levels, those with higher education degrees typically earn more.

The U.S. Census Bureau data from 2004 highlights this trend. Basic high school graduates earned a median income of approximately \(\$37,000\) annually. However, additional years spent in college result in higher median incomes. For instance, obtaining an associate degree (which typically requires two years of college) leads to a noticeable increase. As illustrated in our example, the formula \(I = 37 \times 1.14^y\) helps calculate such income growth. For a bachelor's degree, which takes four years, the income further increases.

Overall, the more educated an individual becomes, the higher their potential earnings. This pattern underscores the significant impact educational attainment has on financial outcomes.

Income Growth Rate

Income growth rate in this context refers to the percentage increase in median income resulting from additional years of education.

In our example, each year of education beyond high school boosts income by \(14\%\). This constant growth rate is crucial for calculating predicted income values using the formula \(I = 37 \times 1.14^y\). Whether pursuing an associate, bachelor’s, or master's degree, the \(14\%\) increase applies.

However, real-world conditions may vary. As observed, the \(14\%\) rate does not apply for Ph.D. students, whose actual median income in 2004 was \(\\(100,000\) instead of the projected \(\\)126,500\). This discrepancy suggests that while the growth rate is a useful approximation for certain degrees, its accuracy might decrease for advanced studies.

Mathematical Modeling in Education

Mathematical modeling helps predict outcomes and make informed decisions. In education, it offers valuable insights into income expectations based on years of schooling.

Using the mathematical function \(I = 37 \times 1.14^y\), the model estimates median incomes based on completed college years. This model provides a systematic approach to visualize potential financial benefits of higher education. It is particularly useful for students planning their educational pathways, enabling them to weigh future income against current educational investment.

While the model is a powerful tool, it must be noted that real-world deviations can occur, as seen with the earnings of Ph.D. holders. Nonetheless, mathematical models are essential in shaping reasonable expectations and driving educational policy decisions.