Our online prediction interval calculator determines the confidence interval for the value of the dependent and independent variable of a given value.
"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
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.
Independent and dependent variables
Confidence Interval
X value for prediction X0
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.
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:
Given Data:
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 |
$$ SS_{XX} = \sum^n_{i=1}X_i^2 - \dfrac{1}{n} \left(\sum^n_{i=1}X_i \right)^2 $$
$$ = 1871 - \dfrac{1}{10} (125)^2 $$
$$ = 308.5 $$
$$ SS_{YY} = \sum^n_{i=1}Y_i^2 - \dfrac{1}{n} \left(\sum^n_{i=1}Y_i \right)^2 $$
$$ = 1931 - \dfrac{1}{10} (131)^2 $$
$$ = 214.9 $$
$$ SS_{XY} = \sum^n_{i=1}{X_iY_i} - \dfrac{1}{n} \left(\sum^n_{i=1}X_i \right) \left(\sum^n_{i=1}Y_i \right) $$
$$ = 1556 - \dfrac{1}{10} (125) (131) $$
$$ = -81.5 $$
$$ \hat{\beta_1} = \dfrac{SS_{XY}}{SS_{XX}} $$
$$ = \dfrac{-81.5}{308.5} $$
$$ = -0.2642 $$
$$ \hat{\beta_0} = \bar{Y} - \hat{\beta_1} \times \bar{X} $$
$$ = 13.1 - -0.2642 \times 12.5 $$
$$ = 16.4025 $$
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:
$$ \hat{Y} = 16.4025 + -0.2642 \times 3 $$
$$ = 15.6099 $$
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:
$$ SS_{Total} = SS_{Regression} + SS_{Error} $$
$$ SS_{Error} = SS_{Total} - SS_{Regression} $$
$$ = 214.9 - 21.5323 $$
$$ = 193.3677 $$
So, mean squared error is:
$$ MSE = \dfrac{SS_{Error}}{n - 2} $$
$$ = \dfrac{193.3677}{10 - 2} $$
$$ = 24.171 $$
$$ \hat{\sigma} = \sqrt{MSE} $$
$$ = \sqrt{24.171} $$
$$ = 4.9164 $$
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
$$ PI = \left(\hat{Y} + E , \hat{Y} - E \right) $$
$$ PI = \left(15.6099 + 12.5316 , 15.6099 - 12.5316 \right) $$
$$ PI = \left(3.0783 , 28.1415 \right) $$
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.
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.
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.
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