Lesson 9

p-Values for t Tests

Big question

What does a p-value measure?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain p-values for t tests in plain language.
  • Use significance level correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

A p-value is the tail probability of a test statistic at least as extreme as the observed one, assuming the null is true.

Key terms

significance level
The chosen probability of rejecting a true null hypothesis.
degrees of freedom
The sample information left after estimating model parameters.
restricted model
The model estimated after imposing the null restrictions.

Core formula

p−value=tailareaunderthenulldistributionp-value = tail area under the null distribution

Use plain-language interpretation before algebra.

Example

GPA1 gives students a real-data setting for p-values for t tests. The lesson emphasizes inference mechanics and interpretation, not memorized output.

Interactive visual

PValueVisualizer

Original Module 4 visual for p-Values for t Tests.

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

p-Values for t Tests Python example

p-Values for t Tests 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

GPA1

Estimated time

25 to 40 min

Packages

pandas, numpy, scipy

Expected output

Printed Python results that can be compared with the lesson explanation.

Learning goals

  • Load and inspect GPA1.
  • Run the Python cells connected to p-Values for t Tests.
  • Interpret the output using p-values and critical values.

Common errors

  • File not found: check that GPA1.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/GPA1.DTA")
df.head()

Interactive activity

PValueVisualizer

Visualize the p-value

Connect the t statistic to tail area and significance levels.

GPA1

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 p-Values for t Tests?

Quick quiz

Which reporting habit is most important in p-Values for t Tests?

Quick quiz

Why is CEOSAL1 a reasonable practice dataset here?

Key takeaway

p-Values for t Tests turns regression output into evidence only when the hypothesis, assumptions, and magnitude are stated clearly.

Download dataDownload resultsDownload interpretation