To find the regression line for the given data, we need to use the formula for the linear regression equation, which is typically written as:

$y = mx + b$

where:

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

We can calculate the slope $m$ and intercept $b$ using the following formulas:

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

2. Intercept (b): $b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2}$

Where:

• $n$ is the number of data points (in this case, $n = 5$).
• $\sum x$ 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.

Step-by-step Calculation:

Given data points:

• $x = [2, 4, 6, 8, 10]$
• $y = [6, 11, 23, 20, 22]$

1. Sum Calculations:

$\sum x = 2 + 4 + 6 + 8 + 10 = 30$ $\sum y = 6 + 11 + 23 + 20 + 22 = 82$ $\sum x^2 = 2^2 + 4^2 + 6^2 + 8^2 + 10^2 = 4 + 16 + 36 + 64 + 100 = 220$ $\sum xy = (2 \times 6) + (4 \times 11) + (6 \times 23) + (8 \times 20) + (10 \times 22) = 12 + 44 + 138 + 160 + 220 = 574$

2. Calculate Slope (m):

$m = \frac{5(574) - (30)(82)}{5(220) - (30)^2} = \frac{2870 - 2460}{1100 - 900} = \frac{410}{200} = 2.05$

3. Calculate Intercept (b):

$b = \frac{(82)(220) - (30)(574)}{5(220) - (30)^2} = \frac{18040 - 17220}{1100 - 900} = \frac{820}{200} = 4.1$

4. Regression Line Equation:

$y = 2.05x + 4.1$

So, the regression line is: $y = 2.05x + 4.1$

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How can you calculate the correlation coefficient from the regression line data?
2. What does the slope in the regression line represent in the context of this data set?
3. How would the regression line change if you had an additional data point?
4. How can you predict new values of $y$ using this regression equation?
5. Can you explain why we subtract the squares of sums from the sum of squares in the formula?

Tip: When interpreting the slope of the regression line, remember that it represents the change in $y$ for each unit increase in $x$.