To calculate the equation of the least squares regression line, we need to find the slope ($b$) and the intercept ($a$) for the line $\hat{y} = a + bx$. This can be done using the formulas for linear regression:

$b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

$a = \frac{\sum y - b \sum x}{n}$

Where:

• $n$ is the number of data points
• $x$ and $y$ are the values for rock surface area and algae colony density, respectively.

Let's calculate these values.

First, we'll compute the necessary summations: $\sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, and $\sum xy$.

Given dataTo calculate the equation of the least squares regression line, we need to find the slope ($b$) and the intercept ($a$) for the line $\hat{y} = a + bx$. This can be done using the formulas for linear regression:

$b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

$a = \frac{\sum y - b \sum x}{n}$

Where:

• $n$ is the number of data points
• $x$ and $y$ are the values for rock surface area and algae colony density, respectively.

Let's calculate these values.

First, we'll compute the necessary summations: $\sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, and $\sum xy$.

Given

• $x = [51, 55, 49, 79, 43, 37, 69, 45, 48]$
• $y = [154, 48, 24, 35, 36, 171, 11, 185, 27]$

Now let's calculate those.The equation of the least squares regression line is:

$\hat{y} = 222.3648 - 2.7527x$

This means that the slope ($b$) is $-2.7527$ and the intercept ($a$) is $222.3648$, both rounded to four decimal places.

Would you like further details or clarifications?

Here are some follow-up questions related to this regression analysis:

1. What is the interpretation of the slope in this context?
2. How can the regression line be used to predict algae colony density for a given rock surface area?
3. What is the residual sum of squares (RSS) for this regression line?
4. How can you determine the strength of the relationship between $x$ and $y$?
5. How would removing an outlier affect the regression line?

Tip: The slope of a regression line indicates the change in the dependent variable for a one-unit change in the independent variable.