# Prediction Interval Calculator

## Table of Content

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Our online prediction interval calculator determines the confidence interval for the value of the dependent and independent variable of a given value.

## Concept of Prediction Interval:

"A prediction interval is the range of values which is likely to have a future observation that gives the specified setting of predictors"

The estimated range of values that have a single new observation based on previous data has two main sources of uncertainty that influence the prediction interval. These include:

• Estimated mean

• Random Variance of new observance

## Steps To Use Prediction Interval Calculator:

A prediction interval is used in the regression analysis for the determination of an interval with a fall of future observation. Our best predicted value calculator is a valuable tool for anyone who needs to make predictions about the future.

### What To Enter?

• Independent and dependent variables

• Confidence Interval

• X value for prediction X0

### Result Summary:

• Predicted Value: Get the value that your model predicts for given values

• Prediction Interval: Range of values that is likely to have an actual value of dependent variables.

• Confidence Level: Probability of the dependent variables on which the actual value will fall within the prediction interval.

## How to Calculate Prediction Interval?

Our Prediction Interval Calculator online helps you make better decisions by quantifying the uncertainty in your predictions. It uses the below formula to pursue instant calculations:

Prediction Interval = Predicted value ± Standard error × t-multiplier

Where:

• Prediction Value = Predicted value by the regression model
• Standard Error = the measure of the variability of individual observations around the mean.
• T-multiplier = The value from t-distribution determined by the desired level of confidence and degree of freedom.

## Practical Example:

Given Data:

• Independent variables = 6, 11, 9, 8, 12, 14, 6, 16, 24, 19
• Dependent variable = 9, 11, 15, 17, 23, 7, 16, 14, 11, 8
• Confidence Level = 0.70
• X value for prediction X0 = 3

### Solution:

So, the predicted value of the given data that is available for the determination of regression coefficients with the help of the sample value of the predictor and the response variable:

 Obs. X Y X2i Y2i Xi . Yi 1 6 9 36 81 54 2 11 11 121 121 121 3 9 15 81 225 135 4 8 17 64 289 136 5 12 23 144 529 276 6 14 7 196 49 98 7 6 16 36 256 96 8 16 14 256 196 224 9 24 11 576 121 264 10 19 8 361 64 152 Sum = 125 131 1871 1931 1556

#### Step # 1 _ Get a sum of squares:

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

$$= 1871 - \dfrac{1}{10} (125)^2$$

$$= 308.5$$

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

$$= 1931 - \dfrac{1}{10} (131)^2$$

$$= 214.9$$

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

$$= 1556 - \dfrac{1}{10} (125) (131)$$

$$= -81.5$$

#### Step # 2 _ Find slope and y-intercept:

$$\hat{\beta_1} = \dfrac{SS_{XY}}{SS_{XX}}$$

$$= \dfrac{-81.5}{308.5}$$

$$= -0.2642$$

$$\hat{\beta_0} = \bar{Y} - \hat{\beta_1} \times \bar{X}$$

$$= 13.1 - -0.2642 \times 12.5$$

$$= 16.4025$$

Regression equation is here:

$$\hat{Y} = \hat{\beta_0} + \hat{\beta_1} \times X$$

Now that we have the regression equation, we can compute the predicted value for X=3, by simply plugging in the value of X=3 in the regression equation found above:

$$\hat{Y} = 16.4025 + -0.2642 \times 3$$

$$= 15.6099$$

#### Step # 3 _ Calculate the 7% of the value:

Y = 15.6099 is the 7% predicted interval. To find the standard error of the calculations we need to calculate sum of squared errors and the regression sum of the squares.

As we realize sum of square is:

$$SS_{Total} = SS_{YY} = 214.9$$

Regression sum of square is figure out by the following way:

$$SS_{Regression} = \hat{\beta_1} \times SS_{XY}$$

$$= -0.2642 \times -81.5$$

$$= 21.5323$$

Since we know that:

$$SS_{Total} = SS_{Regression} + SS_{Error}$$

#### Step # 4 _ Evaluate the sum of the squared error:

$$SS_{Error} = SS_{Total} - SS_{Regression}$$

$$= 214.9 - 21.5323$$

$$= 193.3677$$

So, mean squared error is:

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

$$= \dfrac{193.3677}{10 - 2}$$

$$= 24.171$$

#### Step # 5 _ Find the standard error:

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

$$= \sqrt{24.171}$$

$$= 4.9164$$

As, we figured a 7% prediction intervel for the predicted value that is 15.6099, the level that is used equals to 0.3. The critical t-value for df = n − 2 = 10 - 2 = 8 degrees of freedom, and α = 0.3 is t = 2.16.

Now, we are organized to determine the margin error for the prediction interval with this all given information.

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

$$= 2.16 \times \sqrt{24.171 \left(1 + \dfrac{1}{10} + \dfrac{\left( 3 - 12.5 \right)^2} {308.5} \right)}$$

$$= 12.5316$$

So, the predicted value of the 7% prediction interval is Y = 15.6099

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

$$PI = \left(15.6099 + 12.5316 , 15.6099 - 12.5316 \right)$$

$$PI = \left(3.0783 , 28.1415 \right)$$

## FAQs:

### Why Prediction Interval is Less Certain Than Confidence Interval?

The prediction interval is less certain than the confidence interval because both terms are used to quantify the level of uncertainty in the given set of data. The reason behind the priority of Prediction intervals over the confidence interval is that it focuses on future events and the confidence interval focuses on past events.

### What is the Effect of Sample Size on Prediction Interval?

As the sample size increases, the prediction interval becomes narrower. This is because a sample size provides information about the population that allows us to make more precise predictions.

## References:

From the source Wikipedia: Prediction interval, Normal distribution, Non-parametric methods, Contrast with other intervals, Bayesian statistics.

From the source study.com: What is a Prediction Interval? How do you make a prediction interval? Prediction Interval vs Confidence Interval, Prediction Interval Formula.

### Alan Walker

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.