Some types of algae have the potential to cause damage to river ecosystems. The accompanying data on y = algae colony density and x = rock surface area for nine rivers are a subset of data that come from a study.

x	51	55	49	79	43	37	69	45	48

y	154	48	24	35	36	171	11	185	27

A button hyperlink to the SALT program that reads: Use SALT.

(a)

Calculate the equation of the least squares regression line. (Round your answers to four decimal places.)

ŷ = 

 + 

 
 

x

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.

Some types of algae have the potential to cause damage to river ecosystems. The accompanying data on y = algae colony density and x = rock surface area for nine rivers are a subset of data that come from a study.

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

Calculate the equation of the least squares regression line. (Round your answers to four decimal places.)

This problem calculates the least squares regression line for algae colony density based on rock surface area across nine rivers. The solution includes the regression formula, slope, and intercept calculations using real-world data, x = [51, 55, 49, 79, 43, 37, 69, 45, 48] and y = [154, 48, 24, 35, 36, 171, 11, 185, 27]. Ideal for high school students studying algebra and statistics.

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