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A regression line is a straight line that best represents the data points on a scatter plot. It's used to show the relationship between the dependent variable (often represented as \(\bar{y}\)) and the independent variable (represented as \(x\)). In the given exercise, the regression line is described by the equation \(\bar{y} = 3x + b\). This equation indicates that for every one unit increase in \(x\), the value of \(\bar{y}\) increases by 3 units (since 3 is the slope). The point where the regression line crosses the \(y\)-axis is called the y-intercept \(b\).
In our example, the regression equation passes through the point (2,5). This means when \(x = 2\), the value of \(\bar{y}\) is 5. Understanding this helps you substitute values to find unknowns, like the y-intercept.

Sample means are the average values of a set of data. For example, \(\bar{x}\) is the mean of the sample \(x\)-values and \(\bar{y}\) is the mean of the sample \(y\)-values. In the context of regression analysis, these means represent the center of the data distribution for both variables.
When you compute the sample means, you sum up all the individual values of either variable and then divide by the number of data points. In a regression problem like ours, the means help us understand the overall trend and relationship between two sets of data. By substituting \(\bar{x}\) into the regression line equation, you can find \(\bar{y}\), providing key insights into the data's behavior and assisting in predicting future values.

The y-intercept of a regression line is where the line crosses the \(y\)-axis, and is represented by \(b\) in the regression equation \(\bar{y} = 3x + b\). To find the y-intercept, you need a point through which the line passes.
In this exercise, we know the regression line goes through (2,5). We substitute these values into \(\bar{y} = 3x + b\) to find \(b\):
\(\bar{y} = 3x + b\) becomes \(5 = 3(2) + b\). Simplifying this, we get:
\(5 = 6 + b\)
Solving for \(b\), we get \(b = 5 - 6 = -1\).
Now, we update our regression equation to \(\bar{y} = 3x - 1\). This explains how to find the y-intercept using a known point on the regression line.

Solving equations is a fundamental skill in regression analysis. Let's break down the steps using our example:
First, identify what you know. We have \(\bar{y}= 3x + b\) and the point (2, 5).
1. Substitute known values to solve for unknowns. Here, insert \(x = 2\) and \(y = 5\):
\(\bar{y} = 3x + b\) becomes \(5 = 3(2) + b\).
2. Simplify the equation: \(5 = 6 + b\).
3. Isolate the unknown (b): \(b = 5 - 6 = -1\).
Finally, substitute the value of \(b\) back into the regression equation to get: \( \bar{y} = 3x - 1\).
Consistently follow these steps: identify knowns, substitute values, simplify, and isolate the unknown to find a solution. In conclusion, equation solving is about maintaining clear and ordered steps.