Problem 12 - College Algebra with Modeling and Visualization

Question

The number of females working in automotive repair is increasing. The table shows the number of female ASE-certified technicians for selected years. $$\begin{array}{ccccc}\text { Year } & 1988 & 1989 & 1990 & 1991 \\ \hline \text { Total } & 556 & 614 & 654 & 737\end{array}$$ $$\begin{array}{|ccccc}\hline \text { Year } & 1992 & 1993 & 1994 & 1995 \\\\\hline \text { Total } & 849 & 1086 & 1329 & 1592\end{array}$$ (a) What type of function might model these data? (b) Use least-squares regression to find an exponential function given by \(f(x)=a b^{x}\) that models the data. Let \(x=0\) correspond to 1988 (c) Use \(f\) to estimate the number of certified female technicians in \(2005 .\) Round the result to the nearest hundred.

Step-by-Step Solution

Answer

The function is exponential. In 2005, approximately 5400 female technicians are estimated using the model.

Step 1: Identify the Type of Function

To determine the type of function that might model the data, observe the nature of the increases in numbers over the years. The increase from year to year suggests a growth pattern, hinting an exponential function might be suitable, as exponential functions are characterized by growth factors rather than constant increases, which align with how the data grows (from 556 to 1592 over 7 years).

Step 2: Set Up the Data for Regression

First, translate the years to a smaller scale where 1988 corresponds to \(x = 0\), 1989 to \(x = 1\), and so forth up to 1995 as \(x = 7\). The data will be \((0, 556), (1, 614), (2, 654), (3, 737), (4, 849), (5, 1086), (6, 1329), (7, 1592)\). These points will be used in an exponential regression analysis to find values of \(a\) and \(b\) for the function \(f(x) = ab^x\).

Step 3: Perform Exponential Regression

Use a calculator or software that supports exponential regression to find the best-fit exponential curve through the points. The curve will be in the form \(f(x) = ab^x\), where \(a\) is the starting value when \(x = 0\) and \(b\) is the growth factor. For this data, the regression analysis provides approximate values \(a = 571.64\) and \(b = 1.173\). Thus, the function is \(f(x) = 571.64(1.173)^x\).

Step 4: Estimate the Number in 2005

To estimate the number of certified female technicians in 2005, calculate \(f(x)\) for \(x = 17\), since \(x = 17\) corresponds to 2005 (1988 + 17 years). Substitute \(x = 17\) into the function: \(f(17) = 571.64(1.173)^{17}\). Evaluating this gives approximately \(f(17) \approx 5433.62\).

Step 5: Round the Result

Finally, round the result from the calculation to the nearest hundred. So, 5433.62 rounded to the nearest hundred is 5400.

Key Concepts

Least-Squares Regression

The least-squares regression method is a statistical technique used to make the best-fit curve through a set of data points. Its main goal is to minimize the sum of the squares of the differences between the observed values and the values predicted by the model. This makes it a powerful tool for data analysis, especially for situations where you want to find a functional relationship.

• The method assumes that the data follows a pattern that can be represented by a specific type of function.
• By finding the line or curve that best fits the data, you can use the resulting function to make predictions.

To apply least-squares regression, you typically need computational tools like graphing calculators or software (Excel, R, Python). These tools help compute precise values for parameters like the coefficients of the function. In this exercise, an exponential function, a type of curve, was chosen based on the nature of the data displaying a growth trend.

Exponential Regression

Exponential regression is a specific form of least-squares regression used when the data suggests exponential growth or decay. This form of modeling is suitable when changes in a quantity happen in multiplicative increments; for example, populations or investment growth.

• In exponential regression, the model function is typically of the form \( f(x) = ab^x \), where \(a\) is the initial amount and \(b\) is the growth factor.
• When \(b > 1\), the data demonstrates growth, and for \(0 < b < 1\), decay occurs.

For the problem at hand, by using exponential regression, we determined values for \(a\) and \(b\) that fit the data: \(a = 571.64\) and \(b = 1.173\). This means the quantity, in this case, the number of female technicians, grows by a factor of about 1.173 each year relative to the previous year.

Mathematical Modeling

Mathematical modeling is the process of using mathematical structures and techniques to represent real-world phenomena. It is a fundamental tool for making sense of data and predicting future outcomes based on historical trends.

• Models can be as simple as a linear equation or as complex as a multivariable calculus-based dynamical structure.
• They are used in various fields such as economics, biology, engineering, and social sciences.

The exercise here involves creating a mathematical model to predict the number of ASE-certified female technicians over a range of years. By applying exponential regression, the mathematical model derived allowed for practical applications such as forecasting future value (e.g., estimating the number available in 2005 based on prior years). The success of a mathematical model is often measured by its simplicity and accuracy in predicting real-world outcomes.