Given the level of X, the prediction interval of a response is always wider than the confidence interval of the mean of the response.Question 5Select one:

The statement is indeed correct.

Explanation

1. Understanding Prediction and Confidence Intervals:

   • A Confidence Interval provides a range of values that is likely to contain the true mean of the population based on sample data. It reflects the uncertainty around the estimate of the population mean.
   • A Prediction Interval, on the other hand, gives a range of values likely to contain a single future observation from the same distribution. Since it accounts not only for the estimation of the mean but also for the variability of individual observations, it ends up being wider.

2. Reasons for Wider Prediction Intervals:

   • Inherent Variability: The prediction interval must account for the variability of individual responses around the mean, which is typically greater than the variability of the mean itself.
   • Additional Error Component: The prediction interval includes an additional error term (the standard deviation) because it predicts individual outcomes according to the underlying data distribution.

Conclusion

Thus, when considering a specific level of $X$, the prediction interval will always be wider than the confidence interval for the mean of the response. This understanding is essential in statistics, particularly when communicating about the uncertainty associated with predictions and estimates.