Regarding simple linear regression, which of the following statements is false?

The statement regarding the impact of the explanatory variable x on the total sum of squares is misleading; hence, it is identified as false. In the context of simple linear regression, the total sum of squares (SST) quantifies the total variability in the response variable (y) around its mean. This total variability can be partitioned into explained variability and unexplained variability.

The relationship established in regression provides insight into how well the explanatory variable (x) accounts for the variability in the response variable (y). Specifically, the total sum of squares is a function of the observed values of y, and while it remains constant regardless of the inclusion of x, the role of x in determining how much of that variability is explained (through the regression line) versus that which remains unexplained (the residuals) is significant.

The other statements hold true within the framework of simple linear regression. The least squares line does indeed pass through the means of both x and y, a characteristic outcome of the least squares method. The squared correlation coefficient being equal to the coefficient of determination reflects the consistency in measuring the strength and direction of the linear relationship between x and y. Lastly, a random pattern in a scatterplot suggesting a low coefficient of determination indicates that the variation