Final answer:
For training sets with billions of samples, stochastic gradient descent or mini-batch gradient descent are the recommended linear regression training algorithms. The least-squares regression line is used for predictions if a linear relationship is apparent, and the slope indicates the correlation strength.
Step-by-step explanation:
When dealing with a training set that includes billions of data samples, it's impractical to use standard batch linear regression algorithms due to their computational and memory restrictions. Instead, you should use stochastic gradient descent (SGD) or mini-batch gradient descent, which are both suitable for large-scale and high-dimensional datasets. These algorithms update the model incrementally and can handle data that won't fit in memory, making them ideal for such large datasets.
Predictions using the least-squares regression line rely on the linearity and distribution of the data. To fit the data with a linear model, it should appear that a line can capture the underlying trend effectively. You'll need to identify any outliers that could distort the results and evaluate whether the least-squares line is valid for predictions, such as estimating the cost of a 300 oz. size laundry detergent.
The slope of the least-squares line represents the predicted change in the dependent variable (e.g., cost) for a one-unit increase in the independent variable (e.g., size). This slope is a crucial element of the model as it informs us about the nature and degree of correlation between variables.