Which statement is considered false regarding the prediction of Y?

In the context of statistical modeling, particularly when predicting a dependent variable Y, understanding the nature of the error term (often denoted as ε) is essential.

The statement that ε is always positive is false because the error term can take on both positive and negative values. This reflects the inherent randomness in prediction; ε represents the difference between the observed value and the predicted value. Thus, it can be positive when the prediction underestimates the actual value, or negative when it overestimates it.

In many statistical models, particularly in linear regression, it is assumed that the error term has a mean of zero. This assumption implies that, on average, the predictions are correct over many observations, leading to an unbiased estimate of the true relationship. This further highlights the inaccuracies in the claim that ε is always positive.

Furthermore, variability of ε is influenced by external factors, which can introduce both reducible errors (bias due to incorrect model assumptions, etc.) that could be minimized with better modeling approaches, and irreducible errors that cannot be controlled or predicted. Understanding these concepts reinforces why ε cannot be strictly positive and helps to clarify its role in the overall prediction of Y.