To find the regression line for the set of points $(5, 5)$, $(7, 7)$, and $(9, 10)$, we need to determine the line of best fit in the form:

$y(x) = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Step 1: Calculate the slope $m$

The formula for the slope $m$ is:

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

Where:

• $n$ is the number of points.
• $\sum x$ is the sum of the $x$-coordinates.
• $\sum y$ is the sum of the $y$-coordinates.
• $\sum xy$ is the sum of the product of corresponding $x$ and $y$ coordinates.
• $\sum x^2$ is the sum of the squares of the $x$-coordinates.

Let's calculate these values:

$\sum x = 5 + 7 + 9 = 21$

$\sum y = 5 + 7 + 10 = 22$

$\sum xy = (5 \times 5) + (7 \times 7) + (9 \times 10) = 25 + 49 + 90 = 164$

$\sum x^2 = 5^2 + 7^2 + 9^2 = 25 + 49 + 81 = 155$

Given $n = 3$:

$m = \frac{3(164) - (21)(22)}{3(155) - (21)^2} = \frac{492 - 462}{465 - 441} = \frac{30}{24} = 1.25$

Step 2: Calculate the y-intercept $b$

The formula for the y-intercept $b$ is:

$b = \frac{\sum y - m(\sum x)}{n}$

$b = \frac{22 - 1.25(21)}{3} = \frac{22 - 26.25}{3} = \frac{-4.25}{3} \approx -1.4167$

Step 3: Write the equation of the regression line

Thus, the regression line is:

$y(x) = 1.25x - 1.4167$

Would you like to explore more details or have any further questions?

Related Questions:

1. How do you interpret the slope of a regression line in a real-world context?
2. What is the correlation coefficient, and how is it related to the regression line?
3. How can you calculate the residuals for each data point based on this regression line?
4. What does the y-intercept represent in the context of a regression line?
5. How would the regression line change if an additional point (11, 12) were added?

Tip: Always visualize the data points and the regression line on a graph to better understand the relationship between variables.