When predicting an individual value of y for a new observation corresponding to a given value of x, which interval should be used?

When predicting an individual value of y for a new observation corresponding to a given value of x in a regression analysis, the appropriate interval to use is the prediction interval. A prediction interval provides a range of values within which we expect a single new observation to fall, taking into account both the variability of the data and the uncertainty in the estimation.

The prediction interval accounts for the inherent variability in individual data points around the predicted regression line. It is wider than a confidence interval, which is used primarily to estimate the average value of y for a given x based on the sample data. Since we are interested in predicting a specific, single value for an individual observation rather than the mean, the prediction interval is necessary as it incorporates this additional variability.

In contrast, options like standard deviation focus on the measure of spread of the data set rather than on the prediction of future observations. The term "magnitude interval" is not a standard statistical term related to prediction, which reinforces that the prediction interval is the correct choice when anticipating the outcome of a new observation in the context of regression analysis.