Lesson 11

Confidence Intervals for Regression Coefficients

Big question

How do intervals summarize uncertainty?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain confidence intervals for regression coefficients in plain language.
  • Use p-value correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

A confidence interval combines an estimate, a critical value, and a standard error to give plausible coefficient values.

Key terms

p-value
The null-tail probability of a statistic at least as extreme as the observed statistic.
linear combination
A weighted sum of regression parameters, such as beta1 minus beta2.
F statistic
A statistic used to test several linear restrictions jointly.

Core formula

betahatj+/−criticalvalue∗se(betahatj)beta_hat_j +/- critical value * se(beta_hat_j)

Use plain-language interpretation before algebra.

Example

WAGE1 gives students a real-data setting for confidence intervals for regression coefficients. The lesson emphasizes inference mechanics and interpretation, not memorized output.

Interactive visual

ConfidenceIntervalBuilder

Original Module 4 visual for Confidence Intervals for Regression Coefficients.

wage_sample.csv

y variable

wage

The dependent variable. It is the outcome students want to explain.

x variable

education

The explanatory variable. It is used to describe changes in wage.

Live Python

Confidence Intervals for Regression Coefficients Python example

Confidence Intervals for Regression Coefficients Python example

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 2Load a Python library needed for data work or regression.
  3. Line 3Load a Python library needed for data work or regression.
  4. Line 5Load the dataset into a pandas DataFrame.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Add an intercept column to the regression design matrix.
  7. Line 8Estimate an ordinary least squares regression.
  8. Line 9Create or update a Python object used in the analysis.
  9. Line 10Create or update a Python object used in the analysis.
  10. Line 11Create or update a Python object used in the analysis.
  11. Line 12Create or update a Python object used in the analysis.
  12. Line 13Create or update a Python object used in the analysis.
  13. Line 14Create or update a Python object used in the analysis.
  14. Line 15Create or update a Python object used in the analysis.
  15. Line 16Create or update a Python object used in the analysis.
  16. Line 17Run this Python instruction as part of the lesson workflow.
  17. Line 18Create or update a Python object used in the analysis.
  18. Line 19Display a result so students can inspect the output.
  19. Line 20Display a result so students can inspect the output.
  20. Line 21Display a result so students can inspect the output.

Python walkthrough

  1. 1Load the real dataset and keep the variables needed for the model.
  2. 2Estimate OLS with statsmodels.
  3. 3Compute the statistic, p-value, or confidence interval.
  4. 4Interpret the result with units, controls, and limitations.

Live notebook

Run this lesson as a notebook

Open an editable notebook cell-by-cell, run Python in the browser, and download the `.ipynb` file for later.

Open live notebookDownload .ipynbDownload .py

Related dataset

WAGE1

Estimated time

35 to 55 min

Packages

pandas, numpy, statsmodels, patsy

Expected output

A regression or inference table with coefficients, uncertainty, and short interpretation notes.

Learning goals

  • Load and inspect WAGE1.
  • Run the Python cells connected to Confidence Intervals for Regression Coefficients.
  • Interpret the output using confidence intervals and WAGE1.

Common errors

  • File not found: check that WAGE1.DTA is installed or use the course data folder.
  • Package import error: use the browser notebook first, then download for local Jupyter if your local packages differ.
  • Column name error: compare your variable names with the dataset variables listed for this notebook.

Dataset path helper

import pandas as pd

df = pd.read_stata("/data/WAGE1.DTA")
df.head()

Interactive activity

ConfidenceIntervalBuilder

Build a confidence interval

Change confidence level and see the interval widen or narrow.

WAGE1

Inputs

Try it yourself

Write one plain-English sentence explaining the main idea from this lesson.

Common mistakes

Check these before you move on.

A regression coefficient describes a pattern unless the assumptions or research design support a causal interpretation.

Quick quiz

What is the safest inference focus in Confidence Intervals for Regression Coefficients?

Quick quiz

Which reporting habit is most important in Confidence Intervals for Regression Coefficients?

Quick quiz

Why is RDCHEM a reasonable practice dataset here?

Key takeaway

Confidence Intervals for Regression Coefficients turns regression output into evidence only when the hypothesis, assumptions, and magnitude are stated clearly.

Download dataDownload resultsDownload interpretation