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# Prediction Interval Calculator

Enter the dependent and independent variable in the calculator and the tool will find the prediction interval of range.

The prediction interval calculator estimates the confidence interval for the independent variable (X) and the dependent variable (Y) of a given set of data values.

## What is the Prediction Interval Statistic?

A prediction interval defines a certain range of values around which the response is going to fall or is expected to fall. For example, for 95 % of the prediction interval or range [5,10], you are 95 % certain that the next value is going to fall in this range.  The indicated prediction interval calculator online makes it clear what is the confidence level of a certain range or a prediction in regression analysis. The confidence interval of the linear regression values of the response variable can be checked by the prediction interval.

### How to Calculate the Prediction Interval?

Consider the data sample of the independent variables 6, 7, 7, 8, 12, 14, 15, 16, 16, 19, and the dependent variable 14, 15, 15, 17, 18, 18, 16, 14, 11, and 8. The confidence level is 95% and the Xo is “3”.

#### Solution:

So, the predicted value of the The given data that is available for dependent and independent variables:

 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

Now by the predicted and the response variable, we construct the following table

 Obs. X Y Xᵢ² Yᵢ² Xᵢ · Yᵢ 1 6 14 36 196 84 2 7 15 49 225 105 3 7 15 49 225 105 4 8 17 64 289 136 5 12 18 144 324 216 6 14 18 196 324 252 7 15 16 225 256 240 8 16 14 256 196 224 9 16 11 256 121 176 10 19 8 361 64 152 Sum = 120 146 1636 2220 1690

The predicted value calculator draws the tables of the dependent and independent variables and evaluates the best-fitting prediction interval.

$$SS_{XX} = \sum^n_{i-1}X_i^2 - \dfrac{1}{n} \left(\sum^n_{i-1}X_i \right)^2$$

$$= 1636 - \dfrac{1}{10} (120)^2$$

$$= 196$$ $$SS_{YY} = \sum^n_{i-1}Y_i^2 - \dfrac{1}{n} \left(\sum^n_{i-1}Y_i \right)^2$$

$$= 2220 - \dfrac{1}{10} (146)^2$$

$$= 88.4$$

$$SS_{XY} = \sum^n_{i-1}X_iY_i - \dfrac{1}{n} \left(\sum^n_{i-1}X_i \right) \left(\sum^n_{i-1}Y_i \right)$$

$$= 1690 - \dfrac{1}{10} (120) (146)$$

$$= -62$$

The slope of the line and the y-intercepts are calculated by the formulas:

$$hat{\beta}_1 = \dfrac{SS_{XY}}{SS_{XX}}$$

$$= \dfrac{-62}{196}$$ $$= -0.31633$$

$$hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \times \bar{X}$$

$$= 14.6 - -0.31633 \times 12$$

$$= 18.396$$

Then, the regression equation is:

$$hat{Y} = 18.396 -0.31633X$$

Now, The total sum of the square is:

$$SS_{Total} = SS_{YY} = 88.4$$

Also, the regression sum of the square is calculated as:

$$SS_{R} = \hat{B}_1 SS_{XY}$$

$$= -0.31633 \times -62$$

$$= 19.612$$

Now:

$$SS_{E} = SS_{Total} - SS_{R}$$

$$= 88.4 - 19.612$$

$$SS_{E} = 68.788$$

So, the mean squared error is:

$$MSE = \dfrac{SS_{Error}}{n - 2}$$

$$= \dfrac{68.7894}{10 - 2}$$

$$= 8.5987$$

By picking the square root we find the standard error:

$$hat{\sigma} = \sqrt{MSE}$$

$$= \sqrt{8.5987}$$

$$= 2.9324$$

As, we figure a 95% prediction interval for the predicted value is 17.4467, and the level that is used equals 0.05 as verified by 95 prediction interval calculator. The critical t-value for df = n − 2 = 10 - 2 = 8 degrees of freedom, and α = 0.05 is t = 2.16.  Now, the data is organized to determine the margin error for the prediction interval with this all given information.

$$E = t_\sigma/2;n-2 \times \sqrt{{\sigma}^2 \left(1 + \dfrac{1}{n} + \dfrac{\left( X_0 - \bar{X} \right)^2} {SS_{XX}} \right)}$$

$$= 2.16 \times \sqrt{8.5987 \left(1 + \dfrac{1}{10} + \dfrac{\left( 3 - 12 \right)^2} {196} \right)} = 7.7916$$

So, the predicted value of the 95% prediction interval is Y = 17.4467

$$PI = \left(\hat{Y} + E , \hat{Y} - E \right$$

$$PI = \left(17.4467 + 7.7916 , 17.4467 - 7.7916 \right$$

$$PI = \left(9.6551 , 25.2383 \right$$

The best predicted value calculator calculates the step-by-step solution of the regression analysis. In this example, you are assured that the 95 % predicted interval fall between the range of (9.6551, 25.2383).

## Working of Prediction Interval Calculator:

Our prediction calculator is quite straightforward to use! It requires a couple of data set values to compute the prediction interval. Let’s see how!

Input:

• Enter the independent(X) and dependent variable(Y)
• Enter the confidence level and Xo values for prediction
• Tap Calculate

Ouput:

• Step-by-step calculation of Prediction Interval
• Range of Prediction Interval

## References:

From the source of the statisticsbyjim.com: Prediction intervals From the source of study.com: Confidence Interval, What is a Prediction Interval?