Use your graphing calculator to find the line of best fit for the given data.
Use the indicated variables and proper function notation in your answer.
Round to 3 decimal places as needed.

1	2	3	4	5	6
851	829	848	902	886	905

To find the line of best fit, we use the method of linear regression. The general form of the line of best fit is given by:

\[ y = mx + b \]

Where:
- \( y \) is the dependent variable,
- \( x \) is the independent variable,
- \( m \) is the slope of the line, and
- \( b \) is the y-intercept.

### Given Data:
- \( x \): 1, 2, 3, 4, 5, 6
- \( y \): 851, 829, 848, 902, 886, 905

### Step 1: Calculate the means of \( x \) and \( y \).
\[
\text{Mean of } x = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5
\]
\[
\text{Mean of } y = \frac{851 + 829 + 848 + 902 + 886 + 905}{6} = \frac{5221}{6} \approx 870.167
\]

### Step 2: Calculate the slope \( m \).
\[
m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]

### Step 3: Calculate the y-intercept \( b \).
\[
b = \bar{y} - m\bar{x}
\]

Now let's compute the values. I'll do this calculation for you.The line of best fit for the given data is:

\[
y = 14.143x + 820.667
\]

This equation represents the linear relationship between \(x\) and \(y\), with the slope \(m = 14.143\) and the y-intercept \(b = 820.667\), both rounded to three decimal places.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How can we calculate the correlation coefficient for this data set?
2. What does the slope of the line of best fit represent in this context?
3. How can we use this equation to predict future values of \(y\) for given \(x\) values?
4. How can we assess the goodness of fit for this linear model?
5. How does changing one data point affect the line of best fit?

**Tip:** Always plot your data points along with the line of best fit to visually assess how well the line represents the data!

Learn how to find the line of best fit using linear regression for the given data set: 851, 829, 848, 902, 886, 905. This problem involves calculating the slope and y-intercept of the line of best fit rounded to three decimal places. Suitable for Grades 9-12 students studying statistics and data analysis.

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