91Ó°ÊÓ

Let \(x=\) day of observation and \(y=\) number of locusts per square meter during a locust infestation in a region of North Africa. $$ \begin{array}{r|rrrrr} \hline x & 2 & 3 & 5 & 8 & 10 \\ \hline y & 2 & 3 & 12 & 125 & 630 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the \((x, y)\) data pairs. Do you think a straight line will be a good fit to these data? Do the \(y\) values almost seem to explode as time goes on? (b) Now consider a transformation \(y^{\prime}=\log y .\) We are using common logarithms of base \(10 .\) Draw a scatter diagram of the \(\left(x, y^{\prime}\right)\) data pairs and compare this diagram with the diagram of part (a). Which graph appears to better fit a straight line? (c) Use a calculator with regression keys to find the linear regression equation for the data pairs \(\left(x, y^{\prime}\right) .\) What is the correlation coefficient? (d) The exponential growth model is \(y=\alpha \beta^{x}\). Estimate \(\alpha\) and \(\beta\) and write the exponential growth equation. Hint: See Problem 22 .

Step by step solution

01

Plot the Original Data

For part (a), plot the points \((x, y)\) on a Cartesian plane. The points are (2, 2), (3, 3), (5, 12), (8, 125), and (10, 630). Upon observing the plot, determine if the data can be well-approximated by a straight line. If the y-values increase rapidly, it may suggest an exponential increase rather than a linear trend.

02

Transform the Data Using Logarithms

Apply the transformation \(y' = \log y\) to the original y-values. Compute each \(y'\) as follows: when \(y = 2\), \(y' = \log(2) = 0.301\); when \(y = 3\), \(y' = \log(3) = 0.477\); when \(y = 12\), \(y' = \log(12) = 1.079\); when \(y = 125\), \(y' = \log(125) = 2.097\); and when \(y = 630\), \(y' = \log(630) = 2.799\).

03

Create Scatter Diagram for Transformed Data

For part (b), plot the points \((x, y')\) where \(x\) remains the same. Compare this new scatter plot with the original plot from Step 1 to determine if the transformed data better approximates a straight line.

04

Find Linear Regression Equation

Using a calculator, input the \((x, y')\) data to perform a linear regression. The regression equation is of the form \(y' = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Also, find the correlation coefficient \(r\) to measure the strength of the linear relationship.

05

Estimate Parameters for Exponential Model

To find \(\alpha\) and \(\beta\) in the model \(y = \alpha \beta^x\), first note that in the linear model \(y' = mx + c\), \(\log(y) = mx + c\). Hence, \(\alpha = 10^c\) and \(\beta = 10^m\). Use the values of \(m\) and \(c\) from the regression equation to calculate \(\alpha\) and \(\beta\).

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Key Concepts

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

Exponential Growth Model

The exponential growth model is a mathematical expression that describes how a quantity grows at a rate proportional to its current value. It is expressed in the form

• \( y = \alpha \beta^x \)

where:

• \( y \) is the quantity of interest, which in this exercise is the number of locusts per square meter.
• \( x \) represents the time or the independent variable, such as days in the observation.
• \( \alpha \) is the initial amount or initial value, essentially the value of \( y \) when \( x = 0 \).
• \( \beta \) is the growth factor or base of the exponential.

Over time, values under the exponential growth model increase rapidly, as depicted for the locust infestation where the number of locusts grows significantly in a short period. To derive the exponential growth equation, you first log-transform the data to align it into a linear relationship, which allows for the estimation of \( \alpha \) and \( \beta \) using linear regression methods. The slope of the regression line represents the logarithm of \( \beta \), and the y-intercept represents the logarithm of \( \alpha \). Converting back from logarithmic values gives the parameters for the exponential model.

Logarithmic Transformation

Logarithmic transformation is a useful technique to handle data that grows or decreases exponentially. When data is transformed using common logarithms, it can convert multiplicative relationships into additive ones, making trends more linear and manageable. In this exercise, we have transformed the original y-values associated with the number of locusts using the transformation:

• \( y' = \log_{10} y \)

where \( y' \) represents the transformed values.

• This transformation helps to normalize the variance, which can stabilize the variance of data points that increase rapidly.
• It simplifies the application of linear regression, making patterns easier to identify and analyze.

After applying the transformation, the relationship between x (days of observation) and \( y' \) became linear, thereby allowing for a better fit when plotted. Logarithmic transformations are particularly powerful in revealing exponential growth trends and are often used in bioinformatics, finance, and other fields where growth rates change rapidly.

Scatter Diagram

A scatter diagram is a graphical representation that shows the relationship between two quantitative variables. In this exercise, scatter diagrams were used to visually analyze how the number of locusts (\( y \)) changes with time (\( x \)).

• Initially, we plot the original data points
• For \((x, y)\): (2, 2), (3, 3), (5, 12), (8, 125), and (10, 630).

This scatter plot showed an upward curve, indicating that data grew exponentially rather than linearly. Subsequently, a scatter plot was created for the transformed data:

• \((x, y')\): (2, 0.301), (3, 0.477), (5, 1.079), (8, 2.097), (10, 2.799).
• This showed a linear pattern, suggesting a clearer line fit.

Scatter diagrams are critical for determining whether a linear model is appropriate for the data. They help in visualizing patterns or trends, detecting anomalies or outliers, and assessing the strength of the relationship between variables, essential for model selection and application.