The equation you've provided is a form of a statistical equation, which often appears in the context of linear regression, where we are trying to solve for parameters of a linear model $y = bx + a$. The equation involves sums of products and squares, typically related to data points.

Let's break down the equation step by step to solve it:

Given: $\frac{\left(\sum xy\right) - b\left(\sum x\right)}{\sum x^2} = \frac{\left(\sum y\right) - b\left(\sum 1\right)}{\sum x}$

Step 1: Cross-multiply to eliminate the denominators

We get: $\left(\left(\sum xy\right) - b\left(\sum x\right)\right) \cdot \left(\sum x\right) = \left(\left(\sum y\right) - b\left(\sum 1\right)\right) \cdot \left(\sum x^2\right)$

Step 2: Expand both sides

Expanding both sides gives us: $\left(\sum xy\right)\left(\sum x\right) - b\left(\sum x\right)^2 = \left(\sum y\right)\left(\sum x^2\right) - b\left(\sum 1\right)\left(\sum x^2\right)$

Step 3: Collect terms involving $b$

We will move all the terms involving $b$ to one side of the equation and the rest to the other side: $b\left(\sum x\right)^2 - b\left(\sum 1\right)\left(\sum x^2\right) = \left(\sum y\right)\left(\sum x^2\right) - \left(\sum xy\right)\left(\sum x\right)$

Step 4: Factor out $b$ on the left-hand side

$b\left[\left(\sum x\right)^2 - \left(\sum 1\right)\left(\sum x^2\right)\right] = \left(\sum y\right)\left(\sum x^2\right) - \left(\sum xy\right)\left(\sum x\right)$

Step 5: Solve for $b$

Finally, divide both sides by the expression in brackets to isolate $b$: $b = \frac{\left(\sum y\right)\left(\sum x^2\right) - \left(\sum xy\right)\left(\sum x\right)}{\left(\sum x\right)^2 - \left(\sum 1\right)\left(\sum x^2\right)}$

This equation represents the slope $b$ in a simple linear regression model, where $\sum 1$ is simply the number of data points $n$.

Summary:

The solution for $b$ is: $b = \frac{\left(\sum y\right)\left(\sum x^2\right) - \left(\sum xy\right)\left(\sum x\right)}{\left(\sum x\right)^2 - n\left(\sum x^2\right)}$

Would you like further details, or do you have any questions?

Related Questions:

1. What is the significance of the slope $b$ in linear regression?
2. How do you calculate the sum of squares $\sum x^2$ for a dataset?
3. What does the term $\sum xy$ represent in statistical analysis?
4. How does the equation change if we consider weighted sums?
5. How would you interpret the case where the denominator equals zero?

Tip:

Understanding the derivation of the formula for the slope in linear regression can deepen your comprehension of how data points influence the final model.