Question

Create a scatter plot of the points $(x,\ln y)$ to determine whether an exponential model fits the data. If so, find an exponential model for the data. $$\begin{array}{|cccccc|}\hline \mathit{x}& -13& -6& 1& 8& 15\\ \mathit{y}& 9.8& 12.2& 15.2& 19& 23.8\\ \hline\end{array}$$

Short answer

First, derive the new y-values by taking the natural log of each of the given y-values. Then, plot the points (x, ln(y)) and if they approximate a straight line, an exponential model can fit the data. Lastly, perform linear regression on these points to find the line of best fit, and use the equation of that line to obtain an exponential model in the form $y=e^{ax+b}$.

Step-by-step solution

Create the scatter plot

To create the scatter plot, first convert the y-values into their corresponding natural logs. The new y-values are ln(9.8), ln(12.2), ln(15.2), ln(19), ln(23.8). Then, plot the points (x, ln(y)) on a cartesian plane.

Determine if an exponential model fits the data

Once the points are plotted, observe if they approximate a straight line. If they do, then an exponential model can fit the data.

Find an exponential model for the data

If an exponential model fits the data, conduct a linear regression analysis to find a line of best fit for the transformed data points. The equation for this line will be in the form y = ax + b. Convert this equation back into exponential form by changing the subject of the formula to $y=e^{ax+b}$. The coefficients a and b will typically come from the linear regression analysis.

Key concepts

Understanding the Scatter Plot

A scatter plot is a visual representation of data points on a Cartesian plane, where each point represents the values of two variables.
In this case, you are asked to create a scatter plot for the transformed data points $x,\ln y$, the natural logarithm of y.
Creating this scatter plot is essential to assess the relationship between the variables and determine if a certain model, like an exponential model, fits your data.

• By plotting the x-values $-13,-6,1,8,15$ against the natural log of y-values, you can see if the data shows any pattern or trend.
• If the plotted points form an approximately straight line, it suggests that the relationship can be modeled exponentially.

Scatter plots are a pivotal first step in visualizing and hypothesizing about potential relationships or models in data. Understanding the relationship visually can guide you towards the right mathematical models.

The Role of the Natural Logarithm

The natural logarithm, denoted as $\ln$, is a mathematical function that helps manage exponential relationships in datasets.
It is particularly useful when you're dealing with exponential growth or decay, as it can transform such patterns into linear ones.
In this example, converting the y-values $9.8,12.2,15.2,19,23.8$ into their natural logarithms simplifies the complexity of the data.

• By using the natural log transformation, we essentially "unpack" the exponential growth, making it easier to assess and model.
• This transformation allows us to apply linear methods—like linear regression—to fit the data.

Utilizing natural logarithms correctly can dramatically simplify solving problems that involve exponential relationships. It is a powerful mathematical tool for data analysis.

Applying Linear Regression

Linear regression is a statistical technique used to find the relationship between two variables by fitting a line to observed data.
When applied to the transformed data points $x,\ln y$, it helps us derive the line of best fit, which is crucial for forming an exponential model.

• The linear regression equation will be of the form $y=ax+b$, where $a$ and $b$ are coefficients determined during the regression analysis.
• These coefficients will illustrate the nature of the relationship between x and $\ln y$.

Once the line is created, it can transform back into an exponential equation by using the formula $y=e^{ax+b}$.
This allows us to describe the original data's relationship in an exponential form. Linear regression is therefore a cornerstone technique in finding the best fitting model for transformed data.

Insights from Data Transformation

Data transformation is a method used to make data more suitable for analysis.
In this exercise, taking the natural log of the y-values is a type of transformation that stabilizes the variance and linearizes relationships.
This transformation enables you to utilize linear regression more effectively, as linear patterns are often easier to interpret and model analytically.

• Transformations can reveal hidden trends and patterns that are not immediately apparent in the raw data.
• They simplify the mathematical processes involved, making complex relationships more manageable.

By transforming your data, subtle details become clearer, leading to more accurate models and conclusions. Whether it’s achieving linearity or handling skewed data, well-thought-out data transformations can significantly enhance your analytical toolkit.