Answer true or false to the following statements and explain your answers. a. In multiple linear regression, we can determine whether we are extrapola

a. In multiple linear regression, we can determine whether we are extrapolating in predicting the value of the response variable for a given set of predictor variable values by determining whether each predictor variable value falls in the range of observed values of that predictor.
b. Irregularly shaped regions of the values of predictor variables are easy to detect with two-dimensional scatterplots of pairs of predictor variables, and thus it is easy to determine whether we are extrapolating when predicting the response variable.
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Step 1
(a)
Identification of extrapolation:
In a multiple linear regression, if the width of the confidence interval of the conditional mean of the response variable for some specified values of the predictor variable is wider, compared to the conditional mean of the response variable for the observed values of the predictor variable, then there is an indication of extrapolation.
The width of the confidence interval of the conditional mean of the response variable for some specified values of the predictor variable is wider, when these values fall outside the range of the observed values of the predictor variables.
Thus, the statement “In a multiple linear regression, we can determine whether we are extrapolating in predicting the value of the response variable for a given set of predictor variable values by determining whether each predictor variable value falls in the range of observed values of that predictor” is True.
Step 2
(b)
Difficulty in detecting extrapolation:
In a multiple linear regression, as the number of predictors increases, it becomes more and more difficult to detect extrapolation, the problem increasing with higher correlation among the predictors and irregularity in the shapes of the observed predictor variable.
When there are multiple predictors, two-dimensional scatterplots of pairs of predictor variables are drawn to detect the relationship between them. Irregularly shaped regions among the predictor variables are more difficult to detect with increasing number of predictors.
Thus, the statement “Irregularly shaped regions of the values of the predictor variables are easy to detect with two-dimensional scatterplots of pairs of predictor variables, and thus it is easy to determine whether we are extrapolating when predicting the response variable” is False.