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Prediction Interval Calculator

Enter the dependent and independent variables in the calculator, and the tool will determine the prediction interval range.

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Prediction Interval Calculator

The prediction interval calculator estimates the confidence interval for a future observation of a dependent variable (Y) given a set of independent variable (X) values. It is especially useful in regression analysis to determine the expected range for a new observation.

What is a Prediction Interval?

A prediction interval defines a range within which a future observation is likely to fall. For example, a 95% prediction interval of [5, 10] means you can be 95% confident that the next observed value will lie between 5 and 10.

Step-by-Step Calculation of Prediction Interval

Example Data:

Independent (X) and dependent (Y) variable observations:

Obs.	X	Y
1	6	14
2	7	15
3	7	15
4	8	17
5	12	18
6	14	18
7	15	16
8	16	14
9	16	11
10	19	8

Step 1: Compute the Standard Error of the Predicted Value

The standard error of a predicted value at a specific \(X_h\) is:

\(SE_{\hat{Y}_h} = \hat{\sigma} \sqrt{1 + \frac{1}{n} + \frac{(X_h - \bar{X})^2}{SS_{XX}}}\)

For example, for \(X_h = 15\):

\(SE_{\hat{Y}_h} = 2.9324 \sqrt{1 + \frac{1}{10} + \frac{(15 - 12)^2}{196}} \)

\(= 2.9324 \sqrt{1 + 0.1 + 0.0459} \)

\(= 2.9324 \sqrt{1.1459} \approx 3.136\)

Step 2: Determine the Critical t-Value

For a 95% confidence level with \(n-2 = 8\) degrees of freedom:

\(t_{0.025,8} \approx 2.306\)

Step 3: Compute the Margin of Error

\(E = t \times SE_{\hat{Y}_h} = 2.306 \times 3.136 \approx 7.232\)

Step 4: Calculate the Prediction Interval

\(\hat{Y}_h \pm E = 13.651 \pm 7.232\)

So, the 95% prediction interval is approximately:

[6.419, 20.883]

Another Example with X₀ = 3

Using the regression calculator, the predicted value is \(\hat{Y} = 17.4467\) and the margin of error \(E = 7.7916\):

\(PI = [\hat{Y} - E, \hat{Y} + E] = [9.6551, 25.2383]\)

How the Prediction Interval Calculator Works

Input:

Enter independent (X) and dependent (Y) variable values.
Specify the confidence level (e.g., 95%) and the X₀ value for prediction.
Click "Calculate".

Output:

Step-by-step calculation of the predicted value and standard error.
The prediction interval range for the given X₀.

References:

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