https://file.aiquickdraw.com/mathbot/file/1728622585532v43wdhwt.jpg

We have data showing how the speed of ants in centimeters per second varies with temperature in degrees Celsius. Let's tackle the questions one by one:

### Part (a) - Find the regression line equation.
You are asked to find the equation for the regression line, giving speed \( S \) as a function of temperature \( t \). The equation of a simple linear regression line has the form:

\[
S(t) = mt + b
\]

Where:
- \( m \) is the slope,
- \( b \) is the y-intercept.

Using the given data points:

\[
\text{Temperature} (t): 25.6, 27.5, 30.3, 30.4, 32.2, 33.0, 33.8
\]
\[
\text{Speed} (S): 2.62, 3.03, 3.57, 3.56, 4.03, 4.17, 4.32
\]

We need to calculate the slope (\( m \)) and intercept (\( b \)) of the regression line. I will do that calculation for you.

### Part (b) - Interpretation of the slope
Once we find the slope \( m \), we interpret it as the change in speed (in centimeters per second) for each degree Celsius increase in temperature.

### Part (c) - Speed at \( t = 29^\circ C \)
Here, we will substitute \( t = 29 \) into the regression equation to estimate the speed \( S \) at 29°C.

### Part (d) - Finding temperature for \( S = 2.46 \) cm/sec
For this part, we will solve for \( t \) when the speed \( S = 2.46 \) cm/sec using the equation of the regression line.

Let me first calculate the equation of the regression line (slope and intercept).The equation of the regression line is:

\[
S(t) = 0.2087t - 2.7295
\]

### Part (b) - Meaning of the slope:
The slope \( m = 0.2087 \) means that for every 1°C increase in temperature, the ants' speed increases by approximately 0.2087 centimeters per second.

### Part (c) - Speed at \( t = 29^\circ C \):
Now, let's calculate the speed at 29°C using the equation \( S(t) = 0.2087t - 2.7295 \).

Let me calculate that next.At \( 29^\circ C \), the speed of the ants is approximately \( 3.32 \) cm/sec.

### Part (d) - Temperature when speed is \( 2.46 \) cm/sec:
To find the temperature at which the speed is \( 2.46 \) cm/sec, we will solve for \( t \) in the equation \( S(t) = 0.2087t - 2.7295 \). I'll solve that now.When the ants are running at a speed of \( 2.46 \) cm/sec, the ambient temperature is approximately \( 24.87^\circ C \).

### Summary:
- **(a)** The regression line equation is \( S(t) = 0.2087t - 2.7295 \).
- **(b)** The slope \( 0.2087 \) means the ants' speed increases by about 0.2087 cm/sec for every 1°C increase in temperature.
- **(c)** At \( 29^\circ C \), the ants run at approximately \( 3.32 \) cm/sec.
- **(d)** When the speed is \( 2.46 \) cm/sec, the temperature is about \( 24.87^\circ C \).

Do you want more details on any of these parts, or have any specific questions?

Here are some related questions you might consider:
1. How would the regression equation change with more data points?
2. Can we estimate the speed of ants at \( 35^\circ C \) using the current model?
3. What does the \( r^2 \)-value of the regression tell us about the model's accuracy?
4. How could non-linear regression models help if the relationship is not perfectly linear?
5. What happens if we extrapolate the model to temperatures much higher than those provided?

**Tip:** When interpreting regression results, always check the residuals to ensure the model fits the data well.

A scientist collected data on the speed at which ants ran at different ambient temperatures. The temperature data (in degrees Celsius) and speed data (in centimeters per second) are given. Tasks include finding a regression line equation, interpreting the slope, calculating ant speed at a specific temperature, and determining the temperature for a specific speed.

Analyze a scientist's data on ant speed based on ambient temperature, using linear regression. Solve for the regression equation, interpret the slope, and calculate predictions for speed at specific temperatures and vice versa.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation