To find the line of best fit for the given data using a graphing utility, we can first use the Least Squares method to determine the equation of the line in the form $y = mx + b$, where:

• $m$ is the slope of the line, and
• $b$ is the y-intercept.

Given the dataTo find the line of best fit for the given data using a graphing utility, we can first use the Least Squares method to determine the equation of the line in the form $y = mx + b$, where:

• $m$ is the slope of the line, and
• $b$ is the y-intercept.

Given the

x & y \\ \hline 2 & 10 \\ 4 & 13 \\ 4 & 12 \\ 6 & 15 \\ 8 & 17 \\ 10 & 19 \\ \end{array}$$ ### Step 1: Find the slope $$m$$ The formula for the slope $$m$$ of the line of best fit is: $$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$$ Where: - $$n$$ is the number of data points, - $$\sum xy$$ is the sum of the products of corresponding $$x$$ and $$y$$ values, - $$\sum x$$ and $$\sum y$$ are the sums of the $$x$$-values and $$y$$-values, respectively, - $$\sum x^2$$ is the sum of the squares of the $$x$$-values. Let's compute these values step by step. #### Step 2: Calculate the necessary sums $$\sum x = 2 + 4 + 4 + 6 + 8 + 10 = 34$$ $$\sum y = 10 + 13 + 12 + 15 + 17 + 19 = 86$$ $$\sum x^2 = 2^2 + 4^2 + 4^2 + 6^2 + 8^2 + 10^2 = 4 + 16 + 16 + 36 + 64 + 100 = 236$$ $$\sum xy = (2 \times 10) + (4 \times 13) + (4 \times 12) + (6 \times 15) + (8 \times 17) + (10 \times 19) = 20 + 52 + 48 + 90 + 136 + 190 = 536$$ #### Step 3: Plug values into the formula for $$m$$ $$m = \frac{6(536) - (34)(86)}{6(236) - (34)^2}$$ $$m = \frac{3216 - 2924}{1416 - 1156} = \frac{292}{260} \approx 1.12$$ ### Step 4: Find the y-intercept $$b$$ The formula for the y-intercept $$b$$ is: $$b = \frac{\sum y - m \sum x}{n}$$ Substituting the values we have: $$b = \frac{86 - 1.12 \times 34}{6} = \frac{86 - 38.08}{6} = \frac{47.92}{6} \approx 7.99$$ ### Final Equation The equation of the line of best fit is approximately: $$y = 1.12x + 7.99$$ Would you like more details on how this was calculated or have any other questions? Here are 5 related questions you can explore: 1. How does changing one data point affect the line of best fit? 2. What is the correlation coefficient, and how can it help assess the fit? 3. How can you calculate the residuals for each data point? 4. What is the difference between interpolation and extrapolation using this line of best fit? 5. How do outliers influence the slope and y-intercept of a line of best fit? **Tip:** Understanding the slope and intercept can help predict values outside the given data range, but be cautious when extrapolating too far!