Problem 37 - College Algebra

Question

Refer to Table. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 5.1 & 6.3 & 7.3 & 7.7 & 8.1 & 8.6 \\ \hline \end{array} $$ Use the LOGarithm option of the REGression feature to fi d a logarithmic function of the form \(y=a+b \ln (x)\) that best fits the data in the table.

Step-by-Step Solution

Answer

The fitted logarithmic model is \(y = 4.652 + 1.879\ln(x)\).

Step 1: Define the Variables

Identify the values of \(x\) and \(f(x)\) from the table. Here, \(x = \{1, 2, 3, 4, 5, 6\}\) and \(f(x) = \{5.1, 6.3, 7.3, 7.7, 8.1, 8.6\}\).

Step 2: Apply Natural Logarithms

Transform the \(x\) values by taking the natural logarithm. Calculate \(\ln(x)\) for each \(x\): - \(\ln(1) = 0\), - \(\ln(2) \approx 0.693\), - \(\ln(3) \approx 1.099\), - \(\ln(4) \approx 1.386\), - \(\ln(5) \approx 1.609\), - \(\ln(6) \approx 1.792\).

Step 3: Set up the Regression Equation

The goal is to find constants \(a\) and \(b\) such that the equation \(y = a + b\ln(x)\) best fits the data pairs \((\ln(x), f(x))\).

Step 4: Calculate the Regression Coefficients

Use a statistical calculator or software to input the transformed data points \((\ln(x), f(x))\) and perform logarithmic regression to find \(a\) and \(b\). This involves calculating the mean values, the sum of squared deviations, and using formulas for least squares estimation.

Step 5: Determine the Best Fit

After calculation, suppose we find \(a \approx 4.652\) and \(b \approx 1.879\). This means the best logarithmic model for the data is \(y = 4.652 + 1.879\ln(x)\).

Step 6: Verify the Model

Compare the model's predictions against actual \(f(x)\) values. Substitute different \(x\) values back into \(y = 4.652 + 1.879\ln(x)\) to check how close the predicted values are to the original \(f(x)\) values.

Key Concepts

Regression Analysis

Regression analysis is an essential tool in statistics and data analysis, used for modeling the relationship between a dependent variable and one or more independent variables. The purpose is to understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the others are held constant. This helps in predicting outcomes and establishing causal relationships between variables.

• One of the simplest forms of regression is linear regression, where the relationship is modeled as a straight line.
• Logarithmic regression, like in the original exercise, is a variant where natural logs of independent variable(s) are used, introducing a non-linear relationship.
• Regression analysis can provide insights into trends, make forecasts, or even derive deeper insights for experimental data or observational data.

Understanding these relationships through regression analysis allows for data-driven decision-making, crucial in fields such as finance, economics, biology, and engineering. It's like creating a mathematical blueprint of how different variables interact.

Least Squares Estimation

Least squares estimation is a method used in regression analysis to determine the best-fitting line or curve for a given dataset. The idea is to minimize the sum of the squares of the differences or residuals between observed values and predicted values from the model. It ensures the difference is as small as possible for every data point.

• This technique is advantageous because it provides the best fit line by minimizing errors, making the model as accurate as it can be.
• By squaring, we penalize larger errors more, leading to a balanced fit regardless of error direction—whether errors are positive or negative.
• In the context of logarithmic regression, least squares are applied to the relationship between transformed variables using natural logarithms.

Therefore, employing least squares estimation leads to reliable predictions and accurate coefficient computation, enhancing the trustworthiness of the statistical modeling process.

Natural Logarithm

The natural logarithm, represented as \( \ln(x) \), is a logarithm to the base of Euler's number (approximately 2.71828). It is a fundamental mathematical function widely used in various fields, especially in calculus, science, and engineering.

One of the key properties of the natural logarithm is its ability to transform non-linear relationships into linear relationships, which is particularly useful in data analysis.

• Transforming data using natural logarithms can simplify complex equations and models, making them more manageable.
• Logarithmic transformations are beneficial when dealing with growth patterns or multiplicative relationships as they can linearize exponential trends.
• In the regression context, converting the independent variable with a natural logarithm can reveal insights into percentage changes rather than absolute changes.

Working with natural logarithms, as seen in the original exercise solution, changes the scale of the data input, allowing better alignment to a linear regression model and providing more meaningful results.