Lecture Notes: Understanding Test Statistics

Introduction

• Beginning of Chapter 11b.
• Focus on reviewing and connecting previous concepts.
• Introduction to "test statistic."

Contextual Example: Bottled Water vs Tap Water

• Discussing the popularity of bottled water despite environmental concerns.
• Key question: Do consumers prefer bottled water over tap water?

Designing an Experiment

• RRCC Components: Randomization, Replication, Control, Compare.
  • Randomization: Randomly assign order of water given to mitigate bias.
  • Replication: Have 100 observational units (pairs of cups).
  • Control: Ensure same cup size, shape, and temperature.
  • Compare: Proportion of preferences to 0.5.
• Include concepts of placebo and blinding.

Introduction to Test Statistic

• A test statistic assesses the strength of evidence against the null hypothesis.
• Test statistic = standardization of observation.
• Utilizes normal distribution for large samples.

Steps to Calculate and Interpret a Test Statistic

• Null Hypothesis (H₀): No difference in preference.
• Alternate Hypothesis (Hₐ): There is a difference.
• Calculations involve:
  • P-hat (sample proportion)
  • Standard error
  • Z-score (test statistic)

Comparing Test Results

• Understand distributions and how sample size affects variability.
• Larger sample size generally means less variability.
• Z-Score Interpretation:
  • Values further from zero indicate stronger evidence against H₀.

Application Example

• Florida students’ taste test: 20 out of 22 preferred bottled.
• Calculate test statistic and interpret:
  • Z = 3.823; indicates bottled preference significantly above hypothesized 50%.

Generalization and Conclusion

• Results from specific samples (e.g., Florida students) may not generalize to other populations.
• Understand assumptions and conditions for using normal distribution in hypothesis testing.
• Key Takeaway: The further the test statistic from zero, the stronger the evidence against H₀.