If the least-squares regression line has slope bv1=5, and two x -values differ by 2, the predicted difference in the x-values is [blank].

3 answers

In the context of least-squares regression, the slope (denoted as \( b_{v1} \) in your case) indicates the change in the predicted value of the dependent variable (usually denoted \( y \)) for a one-unit increase in the independent variable (usually denoted \( x \)).

Given that the slope \( b_{v1} = 5 \), if you have two \( x \)-values that differ by 2, the predicted difference in the \( y \)-values associated with those \( x \)-values can be calculated using the slope of the regression line.

The formula for the predicted difference in the \( y \)-values (\( \Delta y \)) when the \( x \)-values differ by a certain amount (\( \Delta x \)) is:

\[ \Delta y = b_{v1} \times \Delta x \]

Substituting the given values:

\[ \Delta y = 5 \times 2 = 10 \]

Therefore, the predicted difference in the \( y \)-values corresponding to the two \( x \)-values that differ by 2 is 10.

If the correlation coefficient is equal to 0, the slope of the least-squares regression line will be equal to [blank]

If the correlation coefficient (\( r \)) is equal to 0, it indicates that there is no linear relationship between the independent variable (\( x \)) and the dependent variable (\( y \)). Consequently, the slope of the least-squares regression line will also equal 0.

In other words, a slope of 0 means that changes in \( x \) do not predict any changes in \( y \) -- the regression line is horizontal. Thus, when the correlation coefficient is 0, the slope of the least-squares regression line will be equal to 0.