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.†
Temperature	Speed
25.6	2.62
27.5	3.03
30.3	3.57
30.4	3.56
32.2	4.03
33.0	4.17
33.8	4.32
(a) Find the equation of the regression line, giving the speed S as a function of the temperature t. (Round the regression line parameters to four decimal places.)
S(t) = 
 

(b) Explain in practical terms the meaning of the slope of the regression line. (Round your answers to four decimal places.)
The slope of 
 means that for each increase in temperature of one degree Celsius, the ants run 
 centimeters per second 
faster
 

### (a) Find the equation of the regression line.

To find the equation of the regression line, we will use the **least squares method**, which fits a line to the data in the form:

\[
S(t) = a + bt
\]

Where:
- \( S(t) \) is the speed in cm/s (dependent variable),
- \( t \) is the temperature in degrees Celsius (independent variable),
- \( a \) is the intercept, and
- \( b \) is the slope of the regression line.

We need to calculate \( a \) (the intercept) and \( b \) (the slope).

The formulas for \( a \) and \( b \) are as follows:

\[
b = \frac{n \sum (t \cdot S) - \sum t \sum S}{n \sum t^2 - (\sum t)^2}
\]
\[
a = \frac{\sum S - b \sum t}{n}
\]

Where:
- \( n \) is the number of data points,
- \( \sum t \) is the sum of the temperatures,
- \( \sum S \) is the sum of the speeds,
- \( \sum (t \cdot S) \) is the sum of the product of temperature and speed,
- \( \sum t^2 \) is the sum of the squares of the temperatures.

Let me calculate this first.The equation of the regression line is:

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

### (b) Interpretation of the slope
The slope of the regression line, \( b = 0.2087 \), means that for each 1-degree Celsius increase in temperature, the speed of the ants increases by approximately **0.2087 cm/s**.

Would you like further details or clarifications?

Here are 5 related questions to expand on the topic:
1. How is the least squares method used to minimize the error in regression?
2. What would happen to the slope if the data showed no clear trend?
3. How can you assess the goodness of fit for this regression model?
4. Can this regression model predict speeds for temperatures beyond the data range?
5. How do outliers in data affect the slope of a regression line?

**Tip:** Always check if the relationship between variables is linear before applying linear regression, as non-linear patterns may require a different approach.

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.
Temperature	Speed
25.6	2.62
27.5	3.03
30.3	3.57
30.4	3.56
32.2	4.03
33.0	4.17
33.8	4.32
(a) Find the equation of the regression line, giving the speed S as a function of the temperature t. (Round the regression line parameters to four decimal places.)
(b) Explain in practical terms the meaning of the slope of the regression line. (Round your answers to four decimal places.)

Learn how to calculate the equation of the regression line for the speed of ants as a function of temperature. This problem covers key concepts in linear regression and statistics, including the least squares method, slope interpretation, and formula application. Suitable for students in Grades 10-12.

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