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Education The following table gives the percentage for the selected years of the U.S. population with less than 12 years of school and can be found in Glassman \(^{6}\) $$ \begin{array}{|l|ccc|} \hline \text { Year } & 1970 & 1980 & 1985 \\ \hline \text { Percentage } & 47.7 & 33.5 & 26.1 \\ \hline \text { Year } & 1989 & 1990 & 1991 \\ \hline \text { Percentage } & 23.1 & 22.4 & 21.6 \\ \hline \end{array} $$ a. On the basis of this data, find the best-fitting exponential function using exponential regression. Let \(x=0\) correspond to \(1970 .\) Graph. Use this model to estimate the percentage in \(1997 .\) b. Using the model in part (a), estimate when the percentage of the U.S. population with less than year 12 years of school will reach \(15 .\)

Step by step solution

01

Define the Variables

Let the year 1970 correspond to \(x = 0\). This means that for each other year, \(x\) is the number of years since 1970. So, 1980 corresponds to \(x = 10\), 1985 to \(x = 15\), and so forth up to 1991 which corresponds to \(x = 21\). Represent percentages as the dependent variable \(y\). Thus, we have the points: \((0, 47.7), (10, 33.5), (15, 26.1), (19, 23.1), (20, 22.4), (21, 21.6)\).

02

Formulate the Exponential Regression Equation

The exponential function is of the form \(y = a \, b^x\). To use exponential regression, an estimated best-fit line is computed using statistical software or a calculator that supports this function. We calculate constants \(a\) and \(b\) using regression tools, resulting in a specific equation representing the data.

03

Graph the Exponential Function

Using the regression equation identified in the previous step, plot the points \((0, 47.7), (10, 33.5), (15, 26.1), (19, 23.1), (20, 22.4), (21, 21.6)\) alongside the best-fit exponential curve on a graph. Check the plot to look for a good correlation between the model and the empirical data.

04

Estimate the Percentage in 1997

To estimate the percentage for 1997, substitute \(x = 1997 - 1970 = 27\) into the exponential equation. Calculate \(y\) using the result, which gives the estimated percentage of the population in 1997 with less than 12 years of school.

05

Solve for the Year When Percentage Reaches 15

We want to find \(x\) when \(y = 15\). Replace \(y\) in the exponential equation with 15, and solve for \(x\). Use logarithms to simplify the equation: \(15 = a \, b^x\). Thus, \(x = \frac{\log(15/a)}{\log(b)}\). Solve for \(x\) and add 1970 to estimate the year.

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Key Concepts

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

Exponential Function

An exponential function is a mathematical expression commonly used to model growth or decay processes. In our context of education statistics, the exponential function is vital in understanding trends over time, especially when changes become rapid as years progress.
To visualize, an exponential function typically follows the form \( y = a \, b^x \), where:

• \( y \) is the dependent variable representing percentage of the population.
• \( x \) denotes the independent variable, in this case, years since 1970.
• \( a \) signifies the initial amount or percentage at the base year (1970).
• \( b \) is the base of the exponential, showing the growth or decay rate per unit time.

Exponential regressions help find the best-fitting exponential curve through a set of data points, allowing for accurate predictions and analyses.

Data Analysis

Data analysis involves examining, cleansing, transforming, and modeling data to glean useful information. In educational statistics, data points such as years and percentages of educational attainment are crucial.
The given exercise involves:

• Assigning numeric values to specific historical years relative to a starting point (1970). For instance, 1980 becomes \( x = 10 \), 1990 as \( x = 20 \), etc.
• Aligning these values with their corresponding percentages to form coordinate pairs.

Such analysis enables us to apply statistical tools like exponential regression, providing insights into trends, such as the decline in the percentage of the population with less than 12 years of schooling.
It aids in visualizing data trends, making inferential decisions easier.

Educational Attainment

Educational attainment reflects the highest level of education completed by an individual or group. In this context, we are examining the decline in percentages of the U.S. population with less than 12 years of schooling over several years.
The historical data shows a trend of increasing educational attainment over the years:

• In 1970, about 47.7% of the population had less than 12 years of schooling.
• This percentage has significantly decreased by 1991, where only 21.6% lacked higher education attainment.

Understanding these trends helps policymakers and educators gauge the effectiveness of educational initiatives over time. Furthermore, projecting these patterns forward using mathematical models provides potential foresight into future educational attainment.

Mathematical Modeling

Mathematical modeling uses mathematical expressions to represent real-world situations, allowing for analysis and predictions. The exercise requires creating an exponential model to represent the declining trend in educational attainment.
Key tasks in mathematical modeling include:

• Using existing data to establish variables and constants in the exponential equation \( y = a \, b^x \).
• Applying regression analysis to calculate the numerical values of \( a \) and \( b \), best representing the empirical data points.
• Interpreting the model to make future predictions, such as estimating the educational attainment percentage in 1997 or when it will reach a certain threshold, like 15%.

By matching models with real data, we acquire tools to simulate probable futures, framing strategic planning and decision-making.