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Running ants: A scientist collected the following data on the speed, in centimeters per second, at which ants ran at the given ambient temperature, in degrees Celsius. \({ }^{23}\) a. Find the equation of the regression line, giving the speed as a function of the temperature. $$ \begin{array}{|c|c|} \hline \text { Temperature } & \text { Speed } \\ \hline 25.6 & 2.62 \\ \hline 27.5 & 3.03 \\ \hline 30.3 & 3.57 \\ \hline 30.4 & 3.56 \\ \hline 32.2 & 4.03 \\ \hline 33.0 & 4.17 \\ \hline 33.8 & 4.32 \\ \hline \end{array} $$ b. Explain in practical terms the meaning of the slope of the regression line. c. Express, using functional notation, the speed at which the ants run when the ambient temperature is 29 degrees Celsius, and then estimate that value. d. The scientist observes the ants running at a speed of \(2.5\) centimeters per second. What is the ambient temperature?

Step by step solution

01

Calculate the Means

First, we calculate the means of the temperature and speed data. Sum the temperatures and divide by the number of data points to find the mean temperature \( \bar{x} \), and similarly, sum the speeds and divide by the number of data points to find the mean speed \( \bar{y} \).

02

Calculate the Slope (b) of the Regression Line

Using the formula for the slope \( b \), where \( b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \), calculate the individual terms, sum them up, and divide as indicated to find \( b \).

03

Calculate the Y-Intercept (a) of the Regression Line

Use the formula \( a = \bar{y} - b \cdot \bar{x} \), substituting the mean values and the slope calculated in Step 2. This will give you the y-intercept \( a \).

04

Form the Regression Equation

Combine the calculated slope \( b \) and y-intercept \( a \) into the linear equation form \( y = a + bx \).

05

Interpret the Slope

The slope \( b \) represents the change in speed (cm/s) for each additional degree of temperature (°C). It indicates how quickly the speed of ants changes as temperature increases or decreases.

06

Calculate Speed at 29°C

Use the regression equation from Step 4, substitute \( x = 29 \) for the temperature, and solve for \( y \) to estimate the speed at this temperature.

07

Find the Temperature for a Given Speed

To find the temperature when speed is \( 2.5 \) cm/s, use the regression equation from Step 4 and solve for \( x \) by substituting \( y = 2.5 \) cm/s.

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Key Concepts

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

Linear regression equation

In regression analysis, a linear regression equation is a mathematical model that describes the relationship between two variables. In our exercise, we want to understand how the speed at which ants run (dependent variable) is affected by the ambient temperature (independent variable). The linear regression equation is represented as follows:
\[ y = a + bx \]
Here, \(y\) is the predicted speed, \(x\) is the temperature, \(a\) is the y-intercept, and \(b\) is the slope of the line.
Steps to Calculate the Linear Regression Equation:

• Calculate the mean of both temperature and speed data.
• Use the slope formula: \(b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}\) to determine \(b\).
• Calculate the y-intercept using: \(a = \bar{y} - b \cdot \bar{x}\).

These calculations allow us to form the linear regression equation, helping predict the speed of ants at any given temperature based on the provided dataset.

Slope interpretation

Understanding slope interpretation is crucial in analyzing linear regression outcomes. The slope \(b\) in the regression equation indicates the rate of change of the speed of ants per unit of temperature. In other words, it tells us how much the speed of the ants increases or decreases with each degree rise in temperature.
Key Points in Slope Interpretation:

• If \(b > 0\), the relationship is positive, meaning speed increases as temperature rises.
• If \(b < 0\), there is a negative relationship, indicating speed decreases while temperature rises.
• If \(b = 0\), temperature changes do not affect the speed of ants.

In practical terms for our exercise, a positive slope means warmer environments lead to faster speeds in ants. This interpretation provides insights into the biological behavior of ants in varying temperatures.

Temperature and speed relationship

The relationship between temperature and speed in our exercise allows us to explore how environmental factors influence ant behavior.
Analyzing the Relationship:

• A higher temperature generally results in a higher speed, suggesting ants are more active in warmer conditions.
• By using the regression equation, we can predict speeds at temperatures not originally measured.
• This prediction helps understand patterns such as optimal working conditions for ants.

In this context, regression analysis provides a way to quantify the connection between temperature changes and the ants' activity levels. Such information is valuable in ecological studies and practical applications related to insect behavior and environmental planning.

Predictive modeling

Predictive modeling involves using statistical models for forecasting future data based on historical trends. Here, we apply the concept of predictive modeling using our linear regression equation.
Steps in Predictive Modeling for Ants’ Speed:

• Use the regression equation to estimate the speed at a given temperature (e.g., 29°C).
• Substitute the values into the equation: \(y = a + b \cdot x\) where \(x\) is the temperature of interest.
• For unknown temperature, rearrange the equation to solve for \(x\) given a speed \(y\) (e.g., 2.5 cm/s).

Predictive modeling in this exercise allows us to make informed guesses about how ants behave under specific conditions, which can aid in further scientific research or conservation efforts.