Question

what is a residual? what does it mean when a residual is positive?
choose the correct answer below.
a. a residual is the difference between an observed value of the response variable y and the value of the corresponding explanatory variable x. if it is positive, then the response variable is greater than the explanatory variable.
b. a residual is the difference between an observed value of the response variable y and the average value of the response variable. if it is positive, then the response variable is greater than the mean.
c. a residual is the difference between an observed value of the response variable y and the predicted value of y. if it is positive, then the observed value is greater than the predicted value.
d. a residual is a data point that does not fit the pattern of the rest of the data. if it is positive, then the data point should still be included in the data set.

Explanation:

A residual is calculated as \( \text{Residual}=y - \hat{y}\), where \(y\) is the observed value of the response variable and \(\hat{y}\) is the predicted value of the response variable. If \(y-\hat{y}>0\), then \(y>\hat{y}\) (observed value is greater than predicted value).

• Option A: Incorrect. Residual is not related to the difference between response variable \(y\) and explanatory variable \(x\).
• Option B: Incorrect. Residual is not the difference between an observed value of \(y\) and the average value of \(y\).
• Option C: Correct. Follows the formula \( \text{Residual}=y - \hat{y}\).
• Option D: Incorrect. Residual is not defined as a data - point that does not fit the pattern.

Answer:

C. A residual is the difference between an observed value of the response variable y and the predicted value of y. If it is positive, then the observed value is greater than the predicted value.