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The correlation coefficient, often represented as \( r \), can tell us how closely two variables move together. It's a numerical value between -1 and 1. A value close to 1 implies a strong positive linear relationship, while one close to -1 suggests a strong negative linear relationship. If \( r \) is near 0, it means there is little to no linear correlation.
In the context of least squares regression, we calculate \( r \) to understand how well our best-fit line represents our data. To find \( r \) for the data points

• \((0,0)\), \((1,2)\), \((2,1)\)

we use the formula:
\[r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}}\]
Substituting the sums from our dataset, we found \( r = 0.5 \). This indicates a moderate positive correlation between the \( x \) and \( y \) values.

A scatter plot provides a visual representation of the relationship between two variables. It's a graph comprised of points that represent values from two datasets, typically the x-values and y-values. In our exercise, we plot each point like so:

• \((0,0)\), \((1,2)\), and \((2,1)\)

Scatter plots help us visually inspect whether there's a probable correlation between two variables. By examining the pattern of the points, we can detect trends, such as upward or downward slopes. Often, the points might not all lie perfectly along a straight line, but they might suggest a general direction or trend.
In this exercise, these points suggest some positive trend, which is quantified by our correlation coefficient \( r \) of 0.5. This plot lays the groundwork for introducing a best-fit line and analyzing the overall data trend.

The best-fit line, also known as the line of best fit or the regression line, is the straight line that best represents the data on a scatter plot. This line minimizes the sum of the squared differences between the observed values and the values predicted by the line.
The equation of a straight line is given by:

• \( y = mx + b \)

where \( m \) is the slope, and \( b \) is the y-intercept. For our dataset, the best-fit line is formulated as \( y = 0.5x + 0.5 \). This means with each unit increase in \( x \), the average \( y \) increases by 0.5.
On the scatter plot of our data, this line is drawn to show the trend as accurately as possible. It helps us predict \( y \) values for given \( x \) values that aren't in the original data.

To create the best-fit line, calculating the slope \( m \) and intercept \( b \) is essential.

The Slope (\( m \))
The slope of a line indicates its steepness and direction. We use the formula:
\[m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]
For our dataset, with three data points, we calculated:

• \( m = 0.5 \)

This tells us that for every one-unit increase in \( x \), \( y \) will increase by 0.5.

The Intercept (\( b \))
The intercept \( b \) is where the line crosses the y-axis, calculated by:
\[b = \frac{(\sum y) - m(\sum x)}{n}\]
For these specific points, \( b \) also returns a value of 0.5. This shows that when \( x \) is 0, \( y \) is predicted to be 0.5, starting the line's path on the y-axis.
Understanding both the slope and intercept allows us to predict and analyze the behavior of the dataset, giving us deeper insights into the data's linear trend.