Crash Course Statistics - Lecture on Test Statistics

Key Concepts

• General Formula for Test Statistics

  • One formula can be adapted for various statistical tests.
  • Reduces the need to memorize multiple formulas for different scenarios.

• Example Experiment: Best Coffee in Town

  • Situation: Comparing coffee from two popular places, Caf-fiend and The Blend Den.
  • Method: Random sample of 16 friends, half given Caf-fiend coffee and half given The Blend Den coffee.
  • Scores: Caf-fiend (7.6), The Blend Den (7.9).
  • Observed difference: 0.3 points lower for Caf-fiend.

Hypotheses

• Null Hypothesis (H0): No difference between the two coffee shops.
• Alternative Hypothesis (H1): There is a difference between the two coffee shops.
• Equivalent to checking if the difference between means is zero.

Two-Sample T-Test (Independent/Unpaired T-Test)

• Formula Application

  • Test statistic formula adapts to a two-sample t-test.
  • Observed difference: 0.3.
  • Measure of average variation: Standard error accounting for both groups.

• Statistical Significance Testing

  • Choose an alpha level (commonly 0.05).
  • Find critical t-values for the chosen alpha level.
    • For this problem: -2.145 and 2.145 (two-tailed test).
  • Calculate t-statistic: around 0.44.
  • Decision: Fail to reject the null hypothesis as 0.44 lies within -2.145 and 2.145.
  • P-Value Approach: p-value = 0.6684 (also leads to failing to reject H0).
  • Interpretation: High p-value (0.6684) means the observed difference can occur 66.84% of the time under H0.

Paired T-Test

• Design Improvement

  • Consider individual coffee preferences: Pairing people based on coffee affinity.
  • New sample: Same 16 friends, each trying both coffee types (self-matching).

• Hypotheses

  • Null Hypothesis (H0): No difference in ratings between Caf-fiend and The Blend Den.

• Test Execution and Results

  • Observed mean difference: -0.18125.
  • Mean difference expected under H0: 0.
  • T-statistic: -3.212.
  • P-value: 0.00582 (very small).
  • Decision: Reject the null hypothesis.
  • Outcome: From findings, The Blend Den coffee rated slightly higher.

Conclusion

• Statistical Tests & Variability

  • Aim to minimize variability for clearer patterns.
  • Use cases: Testing effects of exercise on sleep, music on hearing.
  • Foundation in comparing observed data to expected results.

• Takeaway

  • Statistical tools aid in designing experiments and hypotheses testing.
  • Understanding and applying t-tests (both two-sample and paired) invaluable for practical scenarios.