A prediction c language defines a certain variety of values round which the reaction goes to fall or is predicted to fall. as an example, for 95 % of the prediction interval or variety [5,10], you're ninety 5 % sure that the subsequent fee goes to fall in this variety. The indicated prediction c language calculator on line makes it clean what is the self assurance stage of a sure range or a prediction in regression evaluation. The self belief c language of the linear regression values of the response variable may be checked thru the prediction c program languageperiod.
Consider the dataset of independent variables: 5, 9, 11, 13, 15, 18, 20, 22, 24, 27, and dependent variables: 10, 12, 15, 17, 19, 23, 25, 28, 30, 34. The confidence level is 95%, and the \( X_0 \) is 10.
Solution:
First, list the observations for the independent (X) and dependent (Y) variables:
| Obs. | X | Y |
| 1 | 5 | 10 |
| 2 | 9 | 12 |
| 3 | 11 | 15 |
| 4 | 13 | 17 |
| 5 | 15 | 19 |
| 6 | 18 | 23 |
| 7 | 20 | 25 |
| 8 | 22 | 28 |
| 9 | 24 | 30 |
| 10 | 27 | 34 |
Now construct the table with additional calculations:
| Obs. | X | Y | Xᵢ² | Yᵢ² | Xᵢ · Yᵢ |
| 1 | 5 | 10 | 25 | 100 | 50 |
| 2 | 9 | 12 | 81 | 144 | 108 |
| 3 | 11 | 15 | 121 | 225 | 165 |
| 4 | 13 | 17 | 169 | 289 | 221 |
| 5 | 15 | 19 | 225 | 361 | 285 |
| 6 | 18 | 23 | 324 | 529 | 414 |
| 7 | 20 | 25 | 400 | 625 | 500 |
| 8 | 22 | 28 | 484 | 784 | 616 |
| 9 | 24 | 30 | 576 | 900 | 720 |
| 10 | 27 | 34 | 729 | 1156 | 918 |
| Sum = | 164 | 213 | 3134 | 5113 | 3052 |
Using these totals:
\(SS_{XX} = 3134 - \dfrac{1}{10} (164)^2 = 452.4\)
\(SS_{YY} = 5113 - \dfrac{1}{10} (213)^2 = 362.1\)
\(SS_{XY} = 3052 - \dfrac{1}{10} (164)(213) = 411.6\)
Slope:
\( \hat{\beta}_1 = \dfrac{SS_{XY}}{SS_{XX}} = \dfrac{411.6}{452.4} = 0.91 \)
Intercept:
\( \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \times \bar{X} = 21.3 - 0.91 \times 16.4 = 6.39 \)
Regression Equation:
\( \hat{Y} = 6.39 + 0.91X \)
95% Prediction Interval for \(X_0 = 10\):
\(PI = \left( \hat{Y} + E, \hat{Y} - E \right)\), where \(E = t \cdot \sqrt{MSE \cdot \left(1 + \dfrac{1}{n} + \dfrac{\left(X_0 - \bar{X}\right)^2}{SS_{XX}} \right)}\).
Follow the same procedure to calculate!
