# How to compute a confidence interval for the expected value of a response variable

## Description

If we have a simple linear regression model, $y = \beta_0 + \beta_1x + \epsilon$, where $\epsilon$ is some random error, then given any $x$ input, $y$ can be veiwed as a random variable because of $\epsilon$. Let’s consider its expected value. How do we construct a confidence interval for that expected value, given a value for the predictor $x$?

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## Using statsmodels and sklearn, in Python

Let’s assume that you already have a linear model. We construct an example one here from some fabricated data. For a review of how this preparatory code works, see how to fit a linear model to two columns of data.

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import statsmodels.api as sm
# Replace the following fake data with your actual data:
xs = [ 34, 9, 78, 60, 22, 45, 83, 59, 25 ]
ys = [ 126, 347, 298, 309, 450, 187, 266, 385, 400 ]
# Create and fit a linear model to the data:
xs = sm.add_constant( xs )
model = sm.OLS( ys, xs ).fit()

Ask the model to do a prediction of one particular input, in this example $x=40$, with a $95\%$ confidence interval included ($\alpha=0.05$). You can replce the $40$ with your chosen $x$ value, or an array of them, and you can replace the $0.05$ with your chosen value of $\alpha$.

(The extra `1`

in the input to `get_prediction`

is a placeholder,
required because the model has been expanded to include a constant term.)

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model.get_prediction( [1,40] ).summary_frame( alpha=0.05 )

mean | mean_se | mean_ci_lower | mean_ci_upper | obs_ci_lower | obs_ci_upper | |
---|---|---|---|---|---|---|

0 | 313.721744 | 36.823483 | 226.648043 | 400.795444 | 45.876725 | 581.566762 |

Our 95% confidence interval is $[226.648, 400.7954]$. We can be 95% confident that the true average value of $y$, given that $x$ is 40, is between 226.648 and 400.7954.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

## Solution, in R

Let’s assume that you already have a linear model. We construct an example one here from some fabricated data.

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# Make the linear model
x <- c(34, 9, 78, 60, 22, 45, 83, 59, 25)
y <- c(126, 347, 298, 309, 450, 187, 266, 385, 400)
model <- lm(y ~ x)

Construct a data frame containing just one entry, the value of the independent variable for which you want to compute the confidence interval.
That data frame can then be passed to R’s `predict`

function to get a confidence interval for the expected value of $y$.

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# Use your chosen value of x below:
data <- data.frame(x=40)
# Compute the confidence interval for y:
predict(model, data, interval="confidence", level=0.95) # or choose a different confidence level; here we use 0.95

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fit lwr upr
1 313.7217 226.648 400.7954

Our 95% confidence interval is $[226.648, 400.7954]$. We can be 95% confident that the true average value of $y$, given that $x$ is 40, is between 226.648 and 400.7954.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

## Topics that include this task

## Opportunities

This website does not yet contain a solution for this task in any of the following software packages.

- Excel
- Julia

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