Based on the data shown below, calculate the regression line (show each value to two decimal places)



2

13.65

3

13.68

4

11.11

5

12.34

6

9.27

7

8


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x: 2,3,4,5,6,7

y: 13.65,13.68,11.11,12.34,9.27,8


 y=  x + 

To calculate the regression line, we need to find the equation of the line in the form:

\[
y = mx + b
\]

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

### Step 1: Find the slope (\( m \))
The formula for the slope is:

\[
m = \frac{n \sum{xy} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}
\]

Where:
- \( n \) is the number of data points (in this case, \( n = 6 \)).
- \( \sum{xy} \) is the sum of the products of corresponding \( x \) and \( y \) values.
- \( \sum{x} \) is the sum of the \( x \) values.
- \( \sum{y} \) is the sum of the \( y \) values.
- \( \sum{x^2} \) is the sum of the squares of the \( x \) values.

### Step 2: Find the y-intercept (\( b \))
The formula for the y-intercept is:

\[
b = \frac{\sum{y} - m \sum{x}}{n}
\]

Now, let's calculate the necessary values.

### Given dataTo calculate the regression line, we need to find the equation of the line in the form:

\[
y = mx + b
\]

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

### Step 1: Find the slope (\( m \))
The formula for the slope is:

\[
m = \frac{n \sum{xy} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}
\]

Where:
- \( n \) is the number of data points (in this case, \( n = 6 \)).
- \( \sum{xy} \) is the sum of the products of corresponding \( x \) and \( y \) values.
- \( \sum{x} \) is the sum of the \( x \) values.
- \( \sum{y} \) is the sum of the \( y \) values.
- \( \sum{x^2} \) is the sum of the squares of the \( x \) values.

### Step 2: Find the y-intercept (\( b \))
The formula for the y-intercept is:

\[
b = \frac{\sum{y} - m \sum{x}}{n}
\]

Now, let's calculate the necessary values.

### Given 
\[
x = [2, 3, 4, 5, 6, 7]
\]
\[
y = [13.65, 13.68, 11.11, 12.34, 9.27, 8]
\]

I'll now calculate these values step-by-step.The equation of the regression line is:

\[
y = -1.15x + 16.52
\]

Here:
- The slope \( m = -1.15 \)
- The y-intercept \( b = 16.52 \)

Would you like further details on any part of the calculation or have any other questions?

Here are 5 related questions to explore further:
1. How would the regression line change if the data set included more points?
2. What is the significance of the slope being negative in this case?
3. How can we calculate the coefficient of determination (\( R^2 \)) for this regression line?
4. Can we predict the \( y \)-value for an \( x \)-value of 8 using this line?
5. How does outlier data affect the regression line?

**Tip:** A negative slope means that as \( x \) increases, \( y \) decreases.

Based on the data shown below, calculate the regression line (show each value to two decimal places)

2

13.65

3

13.68

4

11.11

5

12.34

6

9.27

7

8

x: 2,3,4,5,6,7
y: 13.65,13.68,11.11,12.34,9.27,8

Calculate the linear regression line for the data set (x: 2,3,4,5,6,7 and y: 13.65,13.68,11.11,12.34,9.27,8) with detailed step-by-step formulas and a final regression equation. Suitable for students in Grades 10-12.

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