You use a line of best fit for a set of data to make a prediction about an unknown value. The correlation coefficient for your data set is -1. How confident can you be that your predicted value will be reasonably close to the actual value?
Responses
I can’t be confident at all; this is about as close to a random guess as you can get.
I can’t be confident at all; this is about as close to a random guess as you can get.
I can be a little confident; it might be close, or it might be way off.
I can be a little confident; it might be close, or it might be way off.
I can be very confident; it will be very close, but it probably won’t be exact.
I can be very confident; it will be very close, but it probably won’t be exact.
I can be certain that my predicted value will match the actual value exactly.
The correct response is:
I can be very confident; it will be very close, but it probably won’t be exact.
A correlation coefficient of -1 indicates a perfect negative correlation. This means that for the given data set, every time the independent variable increases or decreases, the dependent variable decreases or increases in a perfectly predictable pattern. The line of best fit will have no deviations from the actual data points, and the prediction made by this line will be as accurate as possible given the linear nature of the relationship.
However, it is important to remember that the correlation describes the relationship within the observed data set. If the prediction is for a value within the range of the observed data (interpolation), you can be very confident that your predicted value will closely match the actual value, assuming the linear relationship continues exactly. If the prediction is for a value outside the range of the observed data (extrapolation), the certainty decreases because you're assuming that the observed perfect linear relationship continues beyond the range of the data, which may not always be the case.