The statement is incorrect. The prediction interval for an individual value of the dependent variable (y) is larger than the confidence interval for the average value of (y) for a given value of the independent variable (x).
Here's why:
Confidence interval: A confidence interval provides a range around the estimated mean value of the dependent variable for a specific value of the independent variable. This range captures the population mean with a certain level of confidence (e.g., 95%).
Prediction interval: A prediction interval predicts the range within which a future individual observation of the dependent variable will likely fall, given the value of the independent variable. Since individual observations can deviate from the mean due to noise and random variations, the prediction interval needs to be wider than the confidence interval to account for this uncertainty.
Therefore, the prediction interval for an individual value of (y) will always be larger than the confidence interval for the average value of (y) for the same value of (x).
In simpler terms, the confidence interval tells you where the average value of (y) is likely to be for a given (x), while the prediction interval tells you where a single, new observation of (y) is likely to fall for that same (x). Since individual observations can vary more than the average, the prediction interval needs to be wider to capture this variability.