When analyzing bivariate data and using a regression line to make predictions about the relationship between two variables, one must be cautious about predictions made for values of the independent variable (X) that are outside the range of the observed data. In the given scenario, the X values observed range from 50 to 100, and corresponding Y values range from 1200 to 2000.
Considering this, let's evaluate each option:
a. "the predicted value for y might turn out to be negative" - While this could theoretically happen if the slope of the regression line was negative and steep enough, it does not inherently disqualify predicting Y when X is -10. This is a case-specific issue and not the general reason why extrapolation is discouraged.
b. "a negative x value cannot be used" - This is not accurate. A negative X value can be used in the context of the regression equation, but in this instance, it is far outside the range of the collected data and thus would be an extrapolation, which is generally unreliable, but not because the X value is negative per se.
c. "the proposed x value is well beyond the range of observed data" - This is the correct choice. Extrapolation involves making predictions outside the range of observed data, which can be unreliable because the relationship indicated by the regression line may not hold true beyond that range. For X values around 50 to 100, we have observed corresponding Y values to determine a pattern or trend. However, at X = -10, such a trend may no longer be valid, and any predictions would be based on an assumption that the observed pattern continues indefinitely, which is not a sound assumption without further context.
d. "the y-intercept might be above zero" - Whether the y-intercept is above zero is irrelevant to the issue of extrapolation. The y-intercept being positive or negative has no bearing on the inappropriateness of using extrapolated X values far outside the observed range for prediction.
Therefore, the correct answer is c. "the proposed x value is well beyond the range of observed data," which makes the use of the regression equation to predict a Y value for an X value of -10 not recommended.
