Question

Explain why the predictor variable is useless as a predictor of the response variable if the slope of the population regression line is $0$.

Short answer

For predicting $y$, the regression equation with $x$ as the single predictor variable is useful.

Step-by-step solution

Given Information

The population regression line has a slope of $0$ (zero).

Explanation

The conditional distribution of the response variable is a normal distribution for each value $x$ of the predictor variable since the slope of the population regression line is ${\beta }_1=0$.

${\beta }_0=\left({\beta }_0+0\times x\right)$ and $\sigma$ are the mean and standard deviation, respectively.

Because $x$ does not appear in either of these two characteristics, it can't be used to forecast $y$.

As a result, the regression equation with $x$ as the only predictor variable can be used to predict $y$.