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Chapter 5: Problem 13

Physiologists have discovered that steady-state oxygen consumption (measured per unit of mass) in a running animal increases linearly with increasing velocity. The slope of this line is called the cost of transport of the animal, since it measures the energy required to move a unit mass 1 unit of distance. Table \(5.10\) gives the weight \(W\), in grams, and the cost of transport \(C\), in milliliters of oxygen per gram per kilometer, of seven animals. \({ }^{35}\) a. Judging on the basis of the table, does the cost of transport generally increase or decrease with increasing weight? Are there any exceptions to this trend? b. Make a plot of \(\ln C\) against \(\ln W\). c. Find a formula for the regression line of \(\ln C\) against \(\ln W\), and add this line to the plot you found in part \(b\). d. The cost of transport for a 20,790-gram emperor penguin is about \(0.43\) milliliter of oxygen per gram per kilometer. Use your plot in part c to compare this with the trend for cost of transport versus weight in the table. Does this confirm the stereotype of penguins as awkward waddlers? e. Find a formula that models \(C\) as a power function of \(W\). $$ \begin{array}{|l|c|c|} \hline \text { Animal } & \text { Weight } W & \begin{array}{c} \text { Cost of } \\ \text { transport } C \end{array} \\ \hline \text { White mouse } & 21 & 2.83 \\ \hline \text { Kangaroo rat } & 41 & 2.01 \\ \hline \text { Kangaroo rat } & 100 & 1.13 \\ \hline \text { Ground squirrel } & 236 & 0.66 \\ \hline \text { White rat } & 384 & 1.09 \\ \hline \text { Dog } & 2600 & 0.34 \\ \hline \text { Dog } & 18,000 & 0.17 \\ \hline \end{array} $$

Short Answer

Expert verified
Cost of transport generally decreases with increasing weight, except for the white rat. Use linear regression on log data for analysis.

Step by step solution

01

Analyze Cost of Transport vs Weight

Examine the data in the table to determine if there is a trend between weight (\(W\)) and cost of transport (\(C\)). Generally, as the weight increases, the cost of transport decreases. The data mostly follows this trend, with the exception of the white rat, which has a higher transport cost than the ground squirrel, despite being heavier.
02

Calculate Natural Logarithms

To plot \( \ln C \) against \( \ln W \), calculate \( \ln \) (natural logarithm) for both \( C \) and \( W \) for each animal in the table. This will transform the data into a linear form for regression analysis.
03

Plot Logarithmic Values

Create a scatter plot with \( \ln C \) on the y-axis and \( \ln W \) on the x-axis using the calculated values from Step 2. The plot will help visualize the relationship and trend between the logarithms of cost of transport and weight.
04

Perform Linear Regression

Using the plotted data from Step 3, perform a linear regression to find the best-fit line. The line will have a formula \( \ln C = a + b \cdot \ln W \), where \( a \) and \( b \) are coefficients determined from the regression analysis. Add this regression line to the plot from Step 3.
05

Compare Emperor Penguin Data

For a 20,790-gram emperor penguin with a cost of transport of 0.43, calculate \( \ln C \) and \( \ln W \) and plot this point on the plot from Step 4. Compare its position relative to the regression line to see if it aligns with the general trend or deviates as an outlier indicating awkward movement.
06

Derive Power Function Formula

The regression formula \( \ln C = a + b \cdot \ln W \) can be translated to a power function \( C = kW^m \) by using the exponentials: \( k = e^a \) and \( m = b \). Use the coefficients from Step 4 to determine \( k \) and \( m \), providing a formula that models the cost of transport as a power function of weight.

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Key Concepts

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

Linear Regression
Linear regression is a statistical method used to examine the relationship between two variables, typically a dependent variable and an independent variable. It aims to model the dependent variable as a function of the independent variable through a linear equation of the form: \[ y = a + bx \] where \( a \) is the y-intercept and \( b \) is the slope of the line, indicating how much \( y \) changes for a unit change in \( x \).
In the context of the exercise, linear regression is applied to determine the relationship between the natural logarithms of the cost of transport \( (\ln C) \) and weight \( (\ln W) \) of various animals. This relationship is crucial for understanding how the cost varies with different weights.
Linear regression in this scenario helps us create a regression line or best-fit line, which is used to predict or explain the dependent variable's behavior based on the independent variable's value. This line is added to a scatter plot of the logarithmic values, allowing for visual analysis of the trend.
Logarithmic Transformation
Logarithmic transformation is a method used in data analysis to transform non-linear relationships into linear ones by taking the natural logarithm of one or both variables.
In this exercise, logarithmic transformation is applied to both weight \( W \) and cost of transport \( C \). By transforming these values into \( \ln W \) and \( \ln C \), the data becomes linearized, allowing us to apply linear regression techniques effectively.
  • Transformed relationships are easier to interpret statistically.
  • Logarithmic transformation is especially useful when dealing with multiplicative relationships.
This transformation is particularly beneficial here because the raw data's original scale is likely exponential or follows a power-law relationship, which makes applying a linear model directly ineffective. Converting the values to their logarithmic forms reveals the linear pattern, facilitating a more straightforward regression analysis.
Cost of Transport
The cost of transport is a vital concept in the study of animal physiology and movement. It refers to the energy expenditure required for an animal to move a unit mass over a unit distance. It is typically measured in terms of oxygen consumption and plays a crucial role in understanding locomotion efficiency.
In the exercise, cost of transport \( C \) is given in milliliters of oxygen per gram per kilometer. This value varies across different species, showing how physiological and structural differences affect energy efficiency while moving.
  • A lower cost indicates more efficient movement, consuming less energy.
  • A higher cost suggests that an animal needs more energy per distance, which might affect its survival and ecological niche.
By examining how the cost changes with weight, one can infer evolutionary adaptations related to size and movement efficiency. Understanding this relationship can help explain why certain animals have evolved to be larger or smaller depending on their environment.
Scatter Plot
A scatter plot is a graphical representation used to depict the relationship between two sets of data points. Each point on a scatter plot represents a pair of values from the datasets, one on the x-axis and the other on the y-axis.
In the current analysis, a scatter plot aids in visualizing the relationship between the logarithms of cost of transport \( \ln C \) and weight \( \ln W \).
Creating a scatter plot helps identify the pattern or trend in the data, making it easier to see whether a linear relationship might exist.
  • Scatter plots also allow for the detection of outliers or deviations from the trend.
  • They provide a visual insight into the correlation strength between the variables.
This visual representation is crucial before applying linear regression, as it suggests whether such a model is suitable and helps validate the linear regression results by showing how well the data clusters around the regression line.

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Most popular questions from this chapter

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