Which of the following is an assumption regarding the residuals in least squares regression?

In least squares regression, one of the key assumptions about the residuals is that they are approximately normally distributed. This assumption is crucial because it allows for valid hypothesis testing and the construction of confidence intervals around the regression coefficients. When residuals are normally distributed, it means that their distribution is bell-shaped, indicating that most of the residuals are close to the mean (which should be zero), and deviations from this mean occur in a predictable manner.

If the residuals are normally distributed, it lends itself well to the application of statistical inference methods, which rely on this characteristic to assess the significance of the regression model and the strength of the relationships it identifies. The normality of residuals is tested using various diagnostic tools, such as Q-Q plots or the Shapiro-Wilk test, which help confirm whether the assumption holds true for a given dataset.

Regarding the other aspects, residuals being dependent would violate the assumption of independence, which is critical in regression modeling. The mean of the residuals should ideally be zero, not some other value, as this indicates that the model does not consistently over or underestimate the actual values. Additionally, while residuals can be positive, they are not required to be so; they can be both positive and negative.