Question

what is a residual? what does it mean when a residual is positive? choose the correct answer below. o a. 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. o 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. o c. 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. o 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 \(y>\hat{y}\).

• Option B is incorrect because the residual is not \(y-\bar{y}\) (where \(\bar{y}\) is the mean of \(y\) values).
• Option C is incorrect because the residual is not \(y - x\) (where \(x\) is the explanatory variable).
• Option D is incorrect because a residual is not just an out - of - pattern data point.

Answer:

A. 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.