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The least-squares predictor is a crucial tool in statistics used to predict the value of a dependent variable based on the values of one or more independent variables. This predictor is represented by the equation \( \hat{y} = a + bx \), where:\
- \( a \) is the intercept of the line, representing the point where the line crosses the y-axis.\
- \( b \) is the slope, indicating how much \( y \) changes for a unit change in \( x \).\

In the context of this exercise, we seek the best fitting line that minimizes the sum of the squared differences between the observed values and the predicted values. This method is known as the "least squares" method. It helps in capturing the trend within the data by finding \( a \) and \( b \) using formulas that incorporate mean values and the correlation between the variables. By using least-squares prediction in our data, we developed an equation \( \hat{y} = \frac{45}{7} - 0.684x \), allowing us to predict \( y \) efficiently based on a given \( x \).

The sum of squares of deviations (SSD) is a methodical way to measure variability or dispersion within a data set. It quantifies how much the values in a data set deviate from the mean of that data set. In other words, it is the sum of the squared differences between each data point and the mean of the data set. The calculations follow: \( SSD_{\bar{y}} = \sum(y - \bar{y})^2 \).\

Another type of SSD is calculated when using a predictor, like the least-squares predictor \( \hat{y} \), represented as \( SSD_{\hat{y}} = \sum(y - \hat{y})^2 \).\

The goal is often to reduce this sum of squares by using a good predictive model. In the given solution, the percentage reduction was calculated by comparing the SSD when using the average \( \bar{y} \) as a predictor versus the least-squares predictor \( \hat{y} \). A significant reduction of 46.86% was achieved, reflecting a more accurate model prediction when applying the least-squares method.

A negative linear relationship depicts a scenario where an increase in one variable leads to a decrease in another. This concept is key when interpreting the correlation coefficient \( r \) in our data example.\

The correlation coefficient \( r \) quantifies the strength and direction of a linear relationship between two variables \( x \) and \( y \). If \( r \) is negative, as calculated in the exercise \( r \approx -0.684 \), it indicates that as \( x \) increases, \( y \) tends to decrease.\

This negative relationship is visible on a scatter plot as data points that trend downwards from left to right, forming a line with a negative slope. Understanding this relationship helps predict future values and understand the nature of the data better. Here, the data demonstrates a clear tendency of \( y \), our dependent variable, to decrease as \( x \), the independent variable, increases.