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Construct the equation for the line of best fit for the following data set.  Draw the line of
best fit and label the intercepts.  Show all of your work in creating the equation using the
intercepts

The image you provided shows a scatter plot with data points. To find the equation for the line of best fit, I'll use a statistical approach, calculating the slope (\(m\)) and y-intercept (\(b\)) based on the given data points. The general equation for a line is:

\[
y = mx + b
\]

### Steps to find the line of best fit:

1. **Identify Data Points**: I will estimate the data points from the graph, as we don't have the exact coordinates. This will allow us to approximate the line of best fit.

2. **Calculate the Slope (\(m\))**: The slope is determined by the change in \(y\) over the change in \(x\). It can be calculated using:

   \[
   m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]

3. **Calculate the y-Intercept (\(b\))**: After finding the slope, use the point-slope form to determine the y-intercept.

   \[
   b = \bar{y} - m\bar{x}
   \]

4. **Draw the Line of Best Fit**: Once the equation is derived, you can draw the line on the scatter plot.

### Estimating Points:

From the graph, let's estimate some data points:

- \((1, 5.5)\)
- \((2, 4.8)\)
- \((3, 4.0)\)
- \((4, 3.2)\)
- \((5, 2.5)\)
- \((6, 1.7)\)
- \((7, 1.0)\)

### Calculations:

#### 1. Calculate the Mean of \(x\) and \(y\):

\[
\bar{x} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{7} = 4
\]
\[
\bar{y} = \frac{5.5 + 4.8 + 4.0 + 3.2 + 2.5 + 1.7 + 1.0}{7} \approx 3.24
\]

#### 2. Calculate the Slope (\(m\)):

\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} 
= \frac{(1-4)(5.5-3.24) + (2-4)(4.8-3.24) + \dots + (7-4)(1.0-3.24)}{(1-4)^2 + (2-4)^2 + \dots + (7-4)^2}
\]

This needs to be calculated step by step.

#### 3. Calculate the y-Intercept (\(b\)):

\[
b = \bar{y} - m\bar{x}
\]

Once these are calculated, the line equation \(y = mx + b\) will be formed.

Let me calculate these values for you.The equation for the line of best fit based on the estimated data points is:

\[
y = -0.76x + 6.27
\]

### Drawing the Line of Best Fit:

1. **Slope** (\(m\)): The slope is approximately \(-0.76\), indicating a downward trend.
2. **y-Intercept** (\(b\)): The y-intercept is approximately \(6.27\), meaning the line crosses the y-axis at this point.

### Labeling Intercepts:
- **y-intercept**: \(y = 6.27\) when \(x = 0\).
- **x-intercept**: Set \(y = 0\) and solve for \(x\).

To find the x-intercept:

\[
0 = -0.76x + 6.27
\]

Solving for \(x\):

\[
x = \frac{6.27}{0.76} \approx 8.25
\]

So, the x-intercept is approximately \(x = 8.25\).

Would you like further details on the calculations, or have any other questions?

### Related Questions:
1. How would changing one of the data points affect the slope and intercept?
2. What is the significance of a negative slope in this context?
3. How can you use this equation to predict values outside the given data range?
4. What methods are there to calculate the line of best fit if more accuracy is needed?
5. How would the line of best fit change if the data points had an upward trend?

### Tip:
Understanding the slope's sign (positive or negative) can help you quickly interpret the overall trend of the data set.

Learn how to construct the equation for the line of best fit using given data points. Step-by-step guide includes calculating slope and y-intercept, and drawing the line on a scatter plot. Suitable for Grades 9-12.

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