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L3-10 · Estimation, intervals, and repeated tests

Estimate a probability and describe the sampling meaning of a confidence interval.

Level 3 icon Level 3 · High-school bridge
Objective. Estimate a probability and describe the sampling meaning of a confidence interval.
Before this lesson. L3-07
Hands-on lab. Chance and repeated trials

Learn

An estimator is a rule that maps data to an estimate. A sample proportion estimates the probability of a binary outcome. It varies between repeated samples. This sampling variation differs from model bias, instrument error, and a change in the population over time.

A confidence interval is produced by a procedure with a stated long-run coverage under its assumptions. A 95 percent procedure covers the fixed parameter in about 95 percent of repeated samples. It does not assign a 95 percent frequentist probability to the parameter after observing one interval.

Testing many hypotheses increases opportunities for false findings. Plan the analysis before inspecting outcomes when possible. Report the number of tested claims and use an appropriate correction or independent confirmation. Statistical significance cannot repair confounding, a wrong measurement model, or an undefined causal target.

Practice

Solution

The estimate is 100/400 = 0.25. The approximate standard error is √(0.25×0.75/400), about 0.0217. Independence and a stable binary probability are required for this formula.

Check your understanding

What can a larger independent sample usually reduce?

Show the answer

C. Sampling variation of an estimate

More independent observations can reduce sampling variation. Systematic measurement errors and causal confounding need separate design or modeling solutions.

Sources

This lesson draws on Illowsky & Dean, 2023Hernan & Robins, 2020.

Next: L4-01 · Structural causal models

References
  1. Illowsky, B., & Dean, S. (2023). Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
  2. Hernan, M., & Robins, J. (2020). Causal Inference: What If. https://miguelhernan.org/whatifbook