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Suppose one has drawn a sample from a normally distributed population. The mean and standard deviation of the population are unknown except insofar as they can be estimated based on the sample. It is desired to predict the next observation. Let n be the sample size; let μ and σ be respectively the unobservable mean and standard deviation of the population. Let X1, ..., Xn, be the sample; let Xn+1 be the future observation to be predicted. Let
Then it is fairly routine to show that
has a Student's t-distribution with n − 1 degrees of freedom. Consequently we have
where A is the 100(1 − (p/2))th percentile of Student's t-distribution with n − 1 degrees of freedom. Therefore the numbers
are the endpoints of a 100p% prediction interval for Xn+1.
- Chatfield, C. (1993) "Calculating Interval Forecasts," Journal of Business and Economic Statistics, 11 121-135.
- Meade, N. and T. Islam (1995) "Prediction Intervals for Growth Curve Forecasts," Journal of Forecasting, 14 413-430.
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