Final answer:
Building a regression model might not be possible when there is no significant linear relationship between variables, if the predictions require extrapolation outside the observed data range, or if lurking variables and other factors are affecting the relationship.
Step-by-step explanation:
It might not be possible to build a regression model to predict a response variable for several reasons. If there is no significant linear relationship between the independent variable (x) and the dependent variable (y), then a linear regression model would not be appropriate. For example, if the correlation coefficient is close to zero, it indicates that there is insufficient evidence of a linear relationship, and hence, a linear regression model would not provide reliable predictions.
In situations where the data does not cover the full spectrum of possible x values, making predictions outside the observed value range, known as extrapolation, is unreliable.
This issue arises when the x values in the dataset are within a certain range, and a prediction is needed for a value outside of that range. In such cases, the prediction might not hold true because the model does not have information about the behavior of the system outside the observed range.
Furthermore, a regression might fail to capture the nuances of the relationship when there are lurking variables or unaccounted factors that influence the relationship between x and y, or when the model cannot be experimentally verified. In addition, when only a small percentage of the variation in the response variable is explained by the independent variables (as in the given heart rate example), the model likely does not provide a good fit to the data. Lastly, it is important to remember that correlation does not imply causation; even if a regression model is computed, it cannot be used to imply a causal relationship without additional evidence.