Statistical Analysis and Hypothesis Testing

Nov 1, 2024

Lecture Notes

Introduction

  • Previous lecture: QQ plots and parent heights
  • Created a QQ plot of parent heights, noted deviation in tails.
  • T-tests not suitable due to non-normal distribution.

Dealing with Non-Normality

  • No robust statistical methods covered in this course.
  • For actual research: use robust statistical methods.
  • Robust statistics provide accuracy across a range of distributions.
    • Examples of non-robust statistics: mean, standard deviation.

Evaluating Normality

  • Difficult with small sample sizes.
  • Example: generating non-normal data using R.
  • Plotting with QQ plot to illustrate sample size effect.

Exercise

  • Practice with R code: generating random samples and QQ plots.
  • Importance of understanding sample variability and interpretation.

Effect Size and T-Tests

  • Effect: difference between sample mean and null hypothesized mean.
  • Significance vs. practical relevance.
    • Example: iPhone battery life and statistical significance.
  • Cohen's d: standardized effect size in terms of standard deviations.
    • Small (0.2), Medium (0.5), Large (0.8) effects.

Hedges’ g

  • Used for small sample sizes (<50).
  • Corrected form of Cohen’s d.

Assumptions

  • Data should ideally come from a normally distributed population.

Errors in Statistical Testing

  • Type 1 Error: Rejecting a true null hypothesis.
    • Probability: alpha (often 0.05 or 0.01).
  • Type 2 Error: Failing to reject a false null hypothesis.
    • Probability: beta.

Statistical Power

  • Probability of correctly rejecting a null hypothesis: 1 - beta.
  • Important in determining the reliability of test results.

Distribution Analysis

  • Null distribution and alternative hypothesis distribution discussion.
  • Visual representations using bell curves for understanding errors.

Next Steps

  • Continue with type 2 errors and implications in the next session.

Note: This lecture heavily focuses on understanding the nuances of statistical analysis, particularly dealing with normality, effect sizes, and error types in hypothesis testing. Practical R code examples were provided for creating QQ plots and calculating effect sizes.