Find the regression line associated with the set of points. (Round all coefficients to four decimal places.)
(5, 5), (6, 6), (8, 9)
y(x) =

To find the regression line for the given set of points, we need to use the formula for a straight line:

\[
y = mx + b
\]

Where \(m\) is the slope and \(b\) is the y-intercept. The formulas for the slope and intercept in a linear regression are:

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

\[
b = \frac{\sum{y} - m\sum{x}}{n}
\]

Where:
- \(n\) is the number of points,
- \(\sum{x}\) is the sum of the x-values,
- \(\sum{y}\) is the sum of the y-values,
- \(\sum{xy}\) is the sum of the products of x and y,
- \(\sum{x^2}\) is the sum of the squares of the x-values.

Let's calculate the necessary values:

For the points: (5, 5), (6, 6), (8, 9)
- \(n = 3\)
- \(\sum{x} = 5 + 6 + 8 = 19\)
- \(\sum{y} = 5 + 6 + 9 = 20\)
- \(\sum{xy} = (5 \times 5) + (6 \times 6) + (8 \times 9) = 25 + 36 + 72 = 133\)
- \(\sum{x^2} = 5^2 + 6^2 + 8^2 = 25 + 36 + 64 = 125\)

Now we can compute the slope \(m\) and intercept \(b\).

\[
m = \frac{3(133) - (19)(20)}{3(125) - (19)^2}
\]
\[
b = \frac{20 - m(19)}{3}
\]

I'll calculate these values for you.The equation of the regression line for the given set of points is:

\[
y(x) = 1.3571x - 1.9286
\]

Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How is the slope of a regression line affected by the data points?
2. What is the significance of the y-intercept in a regression line?
3. How would the regression line change if we added more points to the dataset?
4. What assumptions do we make when using linear regression?
5. How can we measure how well the regression line fits the data?

**Tip:** When performing linear regression, the slope represents the change in \(y\) for a unit change in \(x\), and the intercept represents the value of \(y\) when \(x = 0\).

Find the regression line associated with the set of points. (Round all coefficients to four decimal places.) (5, 5), (6, 6), (8, 9) y(x) =

Find the regression line for the points (5, 5), (6, 6), and (8, 9) using linear regression. This detailed solution includes key statistical concepts, the slope and intercept formulas, and step-by-step calculations, suitable for Grades 10-12.

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