y

To find the Line of Best Fit, we will use the least squares method to determine the best linear relationship between the variables $x$ and $y$. The formula for the line of best fit is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Given the data points: $x: 7.2, 2.9, 4.8, 7.1, 3.2, 3.8, 7.6$ $y: 19, 10, 14.1, 19.3, 11.4, 13.6, 21$

Let's calculate the necessary components step-by-step.

1. Calculate the means of $x$ and $y$: $\bar{x} = \frac{7.2 + 2.9 + 4.8 + 7.1 + 3.2 + 3.8 + 7.6}{7}$ $\bar{x} = \frac{36.6}{7}$ $\bar{x} \approx 5.23$

$\bar{y} = \frac{19 + 10 + 14.1 + 19.3 + 11.4 + 13.6 + 21}{7}$ $\bar{y} = \frac{108.4}{7}$ $\bar{y} \approx 15.49$

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

Calculate each term: $(x_i - \bar{x})(y_i - \bar{y})$ $(7.2 - 5.23)(19 - 15.49) = 1.97 \times 3.51 = 6.91$ $(2.9 - 5.23)(10 - 15.49) = -2.33 \times -5.49 = 12.79$ $(4.8 - 5.23)(14.1 - 15.49) = -0.43 \times -1.39 = 0.60$ $(7.1 - 5.23)(19.3 - 15.49) = 1.87 \times 3.81 = 7.12$ $(3.2 - 5.23)(11.4 - 15.49) = -2.03 \times -4.09 = 8.30$ $(3.8 - 5.23)(13.6 - 15.49) = -1.43 \times -1.89 = 2.70$ $(7.6 - 5.23)(21 - 15.49) = 2.37 \times 5.51 = 13.07$

Sum of $(x_i - \bar{x})(y_i - \bar{y})$: $\sum{(x_i - \bar{x})(y_i - \bar{y})} = 6.91 + 12.79 + 0.60 + 7.12 + 8.30 + 2.70 + 13.07$ $\sum{(x_i - \bar{x})(y_i - \bar{y})} = 51.49$

Calculate each term: $(x_i - \bar{x})^2$ $(7.2 - 5.23)^2 = 1.97^2 = 3.88$ $(2.9 - 5.23)^2 = (-2.33)^2 = 5.43$ $(4.8 - 5.23)^2 = (-0.43)^2 = 0.18$ $(7.1 - 5.23)^2 = 1.87^2 = 3.50$ $(3.2 - 5.23)^2 = (-2.03)^2 = 4.12$ $(3.8 - 5.23)^2 = (-1.43)^2 = 2.05$ $(7.6 - 5.23)^2 = 2.37^2 = 5.62$

Sum of $(x_i - \bar{x})^2$: $\sum{(x_i - \bar{x})^2} = 3.88 + 5.43 + 0.18 + 3.50 + 4.12 + 2.05 + 5.62$ $\sum{(x_i - \bar{x})^2} = 24.78$

$m = \frac{51.49}{24.78}$ $m \approx 2.08$

3. Calculate the y-intercept $b$: $b = \bar{y} - m \bar{x}$ $b = 15.49 - (2.08 \times 5.23)$ $b = 15.49 - 10.88$ $b \approx 4.61$

Therefore, the Line of Best Fit is: $y = 2.08x + 4.61$

Would you like more details or have any questions?

Here are five related questions:

1. How would you calculate the correlation coefficient for this data?
2. What is the significance of the slope in the line of best fit?
3. How can we interpret the y-intercept in this context?
4. What are residuals, and how can they be used to assess the fit of the line?
5. How would adding more data points affect the line of best fit?

Tip: Always plot your data and the line of best fit to visually assess the accuracy and fit of the model.