Stat notes confidence

Conceptual Foundation

  • Approach to understanding statistics is conceptual before applying specifics.

  • Transition from investigative scenarios to sampling.

Sampling Basics

  • Starting with a sample to estimate population parameters.

    • Sample mean estimates population mean.

    • Sample proportion estimates population proportion.

    • Sample standard deviation estimates population standard deviation.

Central Limit Theorem (CLT)

  • CLT focuses on sampling variability.

  • Different samples yield different sample results (e.g., different sample means).

Confidence Intervals (CIs)

  • CIs incorporate sample statistics to acknowledge sampling variability.

    • E.g., if a sample indicates 28% attendance, a CI might estimate attendance is between 24% and 32%.

  • Margin of error added to sample statistic to capture uncertainty.

Pop Quiz Exercise

  • Students create CIs for answers about common knowledge questions.

  • Focus is on determining intervals rather than exact answers, using a specified confidence level (90%).

Test Questions for Confidence Intervals

  1. Estimate age with a lower and upper bound.

  2. Length of the Nile River in miles.

  3. Countries in OPEC.

  4. Books in the Old Testament.

  5. Diameter of the moon in miles.

  6. Weight of an empty Boeing 747 in pounds.

  7. Birth year of Mozart.

  8. Gestation period of an Asian elephant in days.

Key Concepts of Confidence Levels

  • Understanding how to quantify confidence in intervals.

  • If truly at 90% confidence, students should correctly identify 9 out of 10 answers by making adequate intervals.

  • Tendency to project more certainty than accuracy.

Psychological Implications

  • Humans often underestimate uncertainty in predictions.

  • Examples of daily life (e.g., arrival times) illustrate this phenomenon.

Economists' Predictions Example

  • Predictions by economists from 1993 to 2010 demonstrate the challenge of accurately predicting outcomes.

  • They missed several years, particularly in booming times or crises (e.g., dot-com boom or 9/11).

Importance of Wider Intervals

  • Wider intervals should be established when uncertainty is high.

  • Goals of confidence intervals should be realistic estimates, rather than unreasonably precise predictions.

Call to Action: Utilization of Mathematical Approaches

  • Use Central Limit Theorem to mathematically express uncertainty and create appropriate confidence intervals.

  • Prepare for further examples on how to compute CIs for means and proportions.