Fandom

Psychology Wiki

Prediction interval

34,202pages on
this wiki
Add New Page
Talk0 Share

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory


In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter.

ExampleEdit

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

\overline{X}_n=(X_1+\cdots+X_n)/n

and

S_n^2={1 \over n-1}\sum_{i=1}^n (X_i-\overline{X}_n)^2.

Then it is fairly routine to show that

{X_{n+1}-\overline{X}_n \over \sqrt{S_n^2+S_n^2/n}} = {X_{n+1}-\overline{X}_n \over S_n\sqrt{1+1/n}}

has a Student's t-distribution with n − 1 degrees of freedom. Consequently we have

P\left(\overline{X}_n-A S_n\sqrt{1+(1/n)}\leq X_{n+1}   \leq\overline{X}_n+A S_n\sqrt{1+(1/n)}\,\right)=p

where A is the 100(1 − (p/2))th percentile of Student's t-distribution with n − 1 degrees of freedom. Therefore the numbers

\overline{X}_n\pm A {S}_n\sqrt{1+(1/n)}

are the endpoints of a 100p% prediction interval for Xn+1.

See alsoEdit

ReferencesEdit

  • 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.
This page uses Creative Commons Licensed content from Wikipedia (view authors).

Also on Fandom

Random Wiki