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].

Question

GPT-4o mini · Answer

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.

Answer

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

GPT-4o mini · Answer

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**.