(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.