Use the technique of linear regression to find the line of best fit for the given points. Round any intermediate calculations to no less than six decimal places, and round the coefficients to two decimal places.
(1, 7), (2, 9), (3, 8), (4, 5), (5, 8), (6, 3), (7, 3)

To find the line of best fit using linear regression, we'll follow these steps:

### Step 1: Calculate the necessary sums
We need the following sums:
- Sum of \(x\) values (\(\sum x\))
- Sum of \(y\) values (\(\sum y\))
- Sum of \(x^2\) values (\(\sum x^2\))
- Sum of \(xy\) values (\(\sum xy\))

Given points: \((1, 7)\), \((2, 9)\), \((3, 8)\), \((4, 5)\), \((5, 8)\), \((6, 3)\), \((7, 3)\)

First, let's calculate these sums.

\[
\sum x = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
\]
\[
\sum y = 7 + 9 + 8 + 5 + 8 + 3 + 3 = 43
\]
\[
\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140
\]
\[
\sum xy = (1)(7) + (2)(9) + (3)(8) + (4)(5) + (5)(8) + (6)(3) + (7)(3) = 7 + 18 + 24 + 20 + 40 + 18 + 21 = 148
\]

### Step 2: Calculate the slope \( m \) and y-intercept \( b \)
The formulas for the slope \(m\) and the y-intercept \(b\) are:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

\[
b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2}
\]

Where \(n\) is the number of data points.

#### Step 2.1: Calculate the slope \( m \)
\[
m = \frac{7(148) - (28)(43)}{7(140) - (28)^2}
\]

Let's break it down:

\[
m = \frac{1036 - 1204}{980 - 784} = \frac{-168}{196} = -0.8571429
\]

Rounded to two decimal places:

\[
m \approx -0.86
\]

#### Step 2.2: Calculate the y-intercept \( b \)
\[
b = \frac{(43)(140) - (28)(148)}{7(140) - (28)^2}
\]

Let's break it down:

\[
b = \frac{6020 - 4144}{196} = \frac{1876}{196} = 9.5714286
\]

Rounded to two decimal places:

\[
b \approx 9.57
\]

### Step 3: Write the equation of the line
The equation of the line of best fit is:

\[
y = mx + b
\]

Substituting the values of \(m\) and \(b\):

\[
y = -0.86x + 9.57
\]

This is the equation of the line of best fit.

Would you like any additional details or have any questions? 

Here are some follow-up questions to consider:
1. How would the line of best fit change if one of the data points was significantly different?
2. What does the slope of the line tell you about the relationship between \(x\) and \(y\)?
3. Can you find the correlation coefficient for this set of points?
4. How would the line of best fit differ if you used a different method like the least squares method?
5. What are the potential limitations of using linear regression for this data set?

**Tip:** When performing linear regression by hand, always carry as many decimal places as possible through your calculations to minimize rounding errors.

Learn how to find the line of best fit using linear regression with a detailed step-by-step solution. This problem involves calculating sums, determining the slope and y-intercept, and writing the equation of the line. Suitable for Grades 11-12 students or those studying advanced statistics.

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