91Ó°ÊÓ

Running speed versus length: The following table gives the length \(L\), in inches, of an animal and its maximum speed \(R\), in feet per second, when it runs. \(^{21}\) (For comparison, 10 feet per second is about \(6.8\) miles per hour.) $$ \begin{array}{|l|c|c|} \hline \text { Animal } & \text { Length } L & \text { Speed } R \\ \hline \text { Deermouse } & 3.5 & 8.2 \\ \hline \text { Chipmunk } & 6.3 & 15.7 \\ \hline \text { Desert crested lizard } & 9.4 & 24.0 \\ \hline \text { Grey squirrel } & 9.8 & 24.9 \\ \hline \text { Red fox } & 24.0 & 65.6 \\ \hline \text { Cheetah } & 47.0 & 95.1 \\ \hline \end{array} $$ a. Does this table support the generalization that larger animals run faster? b. Plot the data points. Does it appear that running speed is approximately a linear function of length? c. Find the equation of the regression line for \(R\) as a function of \(L\), and explain in practical terms the meaning of its slope. (Round regression line parameters to two decimal places.) Add the plot of the regression line to the data plot in part \(b\). d. Judging on the basis of the plot in part c, which is faster for its size, the red fox or the cheetah?

Step by step solution

01

Analyze the Table

Review the table of animal lengths and running speeds. Larger animals like the cheetah have greater speeds, indicating a possible relationship between the size (length) and speed.

02

Determine if Larger Animals Run Faster

In general, larger animals in the given data—like the cheetah—have higher speeds. However, there are smaller animals like the grey squirrel which are faster than longer animals like the desert crested lizard, suggesting that although there is a trend, it is not absolute.

03

Data Plotting

Create a scatter plot with Length ( L ) on the x-axis and Speed ( R ) on the y-axis. Plot each animal as a point.

04

Visualize Linear Relationship

Examine the scatter plot to check if the data points approximately form a straight line, which would suggest that speed is approximately a linear function of length. In this case, the points suggest a somewhat linear relationship but with significant variance especially at higher lengths.

05

Calculate the Regression Line

Using the least squares method, calculate the regression line for speed R as a function of length L . Compute the slope and y-intercept using the formula for a regression line: R = a + bL , where a is the y-intercept and b is the slope.

06

Interpret Regression Line Slope

The slope b of the regression line represents the increase in speed per unit increase in length. It tells us how much speed changes, on average, for each inch increase in the length of the animal.

07

Add Regression Line to Plot

Plot the calculated regression line on the data plot. Observe how well the line fits the data points.

08

Compare Red Fox and Cheetah

In the plot, compare the positions of the red fox and the cheetah relative to the regression line. Determine which one is above the line, indicating it is faster for its size.

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Key Concepts

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

Linear Function

In mathematics, a linear function is a function that creates a straight-line graph. This means that for every unit increase in one variable (the independent variable), there is a constant change in the other variable (the dependent variable). Linear functions can be expressed in the form: \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

In the context of the exercise, we are examining whether running speed is a linear function of the length of animals. If the data points lie approximately along a straight line in a scatter plot, it indicates a linear relationship. This would mean that as the length of the animal increases, its speed increases by a constant factor, showing linear growth.

• Linear functions are characterized by a constant rate of change.
• They can be used to predict one variable based on the other, provided the relationship remains consistent.

Scatter Plot

A scatter plot is a graphical representation of two variables' values, used to determine the type of relationship between them. Each point on a scatter plot represents one observation from a dataset showing two variables' simultaneous values.

For this exercise, you would plot each animal's length (\(L\)) on the x-axis and speed (\(R\)) on the y-axis. This helps visualize whether there's a pattern or trend between length and speed. A clear pattern, especially aligning with a straight line, suggests some relationship, potentially a linear one.

• Scatter plots help identify correlations – whether positive or negative.
• A linear pattern can indicate a potential linear function between the variables.
• Outliers can often be identified, which might skew data interpretation.

Regression Line

A regression line is essentially the best-fitting straight line through a set of data points on a scatter plot. It predicts the value of the dependent variable based on the independent variable. In this exercise, you find the regression line for speed (\(R\)) as a function of length (\(L\)).

To calculate this, we typically use the least squares method that minimizes the sum of the squares of the differences between observed values and those predicted by the line. The regression line's equation can be written as: \(R = a + bL\), where \(a\) is the y-intercept and \(b\) is the slope.

• The slope \(b\) tells us how much \(R\) (speed) changes per unit change in \(L\) (length).
• Points on or near the line demonstrate this relationship.
• This line can help in predicting speeds for animals not part of the dataset, based on their length.

Data Interpretation

Data interpretation involves understanding and extracting meaning from data. Once we have the regression equation, including the scatter plot and the regression line, data interpretation helps answer questions and make informed decisions.

In our problem, interpretation is about seeing whether larger animals generally run faster and understanding which animal is faster for its size. By observing the scatter plot and regression line, you can determine how well the line fits the data points, whether there are any outliers, and compare individual animals like the red fox and the cheetah relative to their expected performance based on length.

• Interpretation helps in validating or refuting hypotheses from the data trends.
• It provides insights into the causative relationships or natural laws at play.
• Comparative analysis allows for deeper extrapolation of data – understanding beyond raw numbers.