To find the line of best fit using linear regression, we need to calculate the slope (m) and y-intercept (b) of the line:

The equation of the line of best fit is:

$y = mx + b$

Where:

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

Steps to find the linear regression line:

1. Calculate the averages:

   • Mean of $x$: $\bar{x}$
   • Mean of $y$: $\bar{y}$

2. Find the slope ($m$): $m = \frac{n \sum{(xy)} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2}$ Where:

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

3. Find the y-intercept ($b$): $b = \bar{y} - m \bar{x}$

Let's calculate the values step-by-step.

Step 1: Calculate the necessary sums

Given data:

• $x = [1, 2, 3, 4, 5, 6]$
• $y = [536, 518, 547, 523, 547, 567]$

We will compute the sums:

• $\sum{x} = 1 + 2 + 3 + 4 + 5 + 6 = 21$
• $\sum{y} = 536 + 518 + 547 + 523 + 547 + 567 = 3238$
• $\sum{x^2} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91$
• $\sum{xy} = (1)(536) + (2)(518) + (3)(547) + (4)(523) + (5)(547) + (6)(567) = 536 + 1036 + 1641 + 2092 + 2735 + 3402 = 9442$

Step 2: Apply the formula for slope $m$

$m = \frac{6(9442) - (21)(3238)}{6(91) - (21)^2}$ $m = \frac{56652 - 67998}{546 - 441}$ $m = \frac{-11346}{105}$ $m \approx -107.8$

Step 3: Apply the formula for intercept $b$

$\bar{x} = \frac{\sum{x}}{n} = \frac{21}{6} = 3.5$ $\bar{y} = \frac{\sum{y}}{n} = \frac{3238}{6} \approx 539.67$ $b = \bar{y} - m\bar{x}$ $b = 539.67 - (-107.8)(3.5)$ $b = 539.67 + 377.3$ $b \approx 916.97$

Final Answer:

The line of best fit is approximately:

$y = -107.8x + 916.97$

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Would you like further explanation on any part of the solution or more questions about linear regression? Here are some related questions you can explore:

1. How would you interpret the slope and intercept in this context?
2. What is the coefficient of determination ($R^2$) and how is it calculated?
3. How can we assess the goodness of fit for the regression model?
4. How would the line of best fit change if you added another data point to the dataset?
5. How would you predict $y$ for $x = 7$ using the equation of the line?

Tip: When interpreting the slope, keep in mind that it shows how much $y$ changes for a one-unit change in $x$.