This is one of the following seven articles on Simple Linear Regression in Excel

Overview of Simple Linear Regression in Excel 2010 and Excel 2013

Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013

Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013

Evaluation of Simple Regression Output For Excel 2010 and Excel 2013

All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013

Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

# Prediction Interval of

Simple Regression in Excel

A prediction interval is a confidence interval about a Y value that is estimated from a regression equation. A regression prediction interval is a value range above and below the Y estimate calculated by the regression equation that would contain the actual value of a sample with, for example, 95 percent certainty.

The Prediction Error for a point estimate of Y is always slightly larger than the Standard Error of the Regression Equation shown in the Excel regression output directly under Adjusted R Square.

The Standard Error of the Regression Equation is used to calculate a confidence interval about the mean Y value. The Prediction Error is use to create a confidence interval about a predicted Y value. There will always be slightly more uncertainty in predicting an individual Y value than in estimating the mean Y value.

For that reason, a Prediction Interval will always be larger than a Confidence Interval for any type of regression analysis.

Calculating an exact prediction interval for any regression with more than one independent variable (multiple regression) involves some pretty heavy-duty matrix algebra. Fortunately a prediction interval for simple regression can be calculated by hand as follows:

## Prediction Interval Formula

For Simple Regression

The formula for a prediction interval about an estimated Y value (a Y value calculated from the regression equation) is found by the following formula:

Prediction Interval = Y_{est} ± t-Value_{α/2,df=n-2} * Prediction Error

Prediction Error = Standard Error of the Regression * SQRT(1 + distance value)

** Distance value**, sometimes called leverage value, is the measure of distance of the combinations of values, x

_{1}, x

_{2},…, x

_{k}from the center of the observed data. Distance value in any type of multiple regression requires some heavy-duty matrix algebra. This is given in Bowerman and O’Connell (1990).

Distance value can be calculated for single-variable regression in a fairly straightforward manner as follows:

Distance value = 1/n + [(x** _{0}** – x_bar)

**]/SS**

^{2}

_{xx}If, for example we wanted to calculate the 95 percent Prediction Interval for the estimated Y value when X = 5000 kg. of input pellets, the following calculations would be performed:

x** _{0}** = 5,000

n = 20

Y_{est} = Number of Parts Produced = 1,345.09 + 1.875 (Weight of Input Pellets in kg.)

Y_{est} = 1,345.09 + 1.875 (5,000)

Y_{est} = 10,730

t-Value_{α/2,df=n-2} = TINV(0.05/2,20-2)

t-Value_{α/2,df=n-2} = TINV(0.975,18) = 2.3987

In Excel 2010 and beyond, TINV(α, n – 2) can also be calculated by the following Excel formula:

TINV(α, n – 2) = T.INV(1-α/2, n -2)

x_bar and SS** _{xx}** are found as follows:

*(Click On Image To See a Larger Version)*

*(Click On Image To See a Larger Version)*

*(Click On Image To See a Larger Version)*

*(Click On Image To See a Larger Version)*

Now we have the following:

x** _{0}** = 5,000

n = 20

Y** _{est}** = 10,730

t-Value** _{α/2,df=n-2}** = 2.3987

x_bar = 2,837.65

SS** _{xx}** = 94,090,690.55

Distance value = 1/n + [(x** _{0}** – x_bar)

**]/SS**

^{2}

_{xx}Distance value = 1/20 + [(5,000 – 2,837)** ^{2}**]/94,090,690

Distance value = 0.099694

Prediction Error = Standard Error of the Regression * SQRT(1 + distance value)

Standard Error of the Regression = 1,400.463

This is found from the Excel regression output as follows:

*(Click On Image To See a Larger Version)*

Prediction Error = 1,400.463 * SQRT(1 + 0.099694)

Prediction Error = 1,400.463 * 1.048663

Prediction Error = 1,468

Prediction Interval = Y** _{est}** ± t-Value

*** Prediction Error**

_{α/2,df=n-2}Prediction Interval = 10,730 ± 2.3987 * 1,468

Prediction Interval = 10,730 ± 3,533

Prediction Interval = [ 7,197, 14,263 ]

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

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- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
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- z-Tests in Excel
- Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013
- One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013
- Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013

- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- 1-Sample t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
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- 2-Independent-Sample Pooled t-Tests in Excel
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- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
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- 2-Sample Pooled t-Test Power With G*Power Utility
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- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
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- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
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- z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression in 6 Steps in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer- Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013
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- ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
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- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
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- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel

- Nonparametric Testing in Excel
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- Post Hoc Testing in Excel
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- Solving Problems With Other Distributions in Excel
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