QQ Plots, Statistical Tests, and Errors

Nov 1, 2024

Lecture Notes: QQ Plots and Statistical Tests

QQ Plots

  • Previous Lecture Review: Discussed QQ plots with parent heights data.
  • Deviation in Tails: Observed deviations in tails of QQ plot indicating non-normality.
  • Non-normality: T-tests not ideal due to non-normality; robust methods needed for real research but beyond course scope.
  • Robust Statistical Methods: Methods that provide accurate info across various distributions; not covering in this course.
  • Impact of Sample Size:
    • Smaller sample sizes make assessing normality harder.
    • Illustrated with generating non-normal data from an exponential distribution.
    • Low sample sizes may misleadingly appear normal.
  • Random Sampling & QQ Plot Practice:
    • Demonstrated writing R code to visualize fluctuations in QQ plots with different sample sizes.
    • Highlighted inevitable sampling errors.

One Sample T-test: Effect Size

  • Concept of Effect: Difference between sample mean and null hypothesized mean.
  • Statistical Significance vs. Meaningfulness:
    • Importance of practical significance alongside statistical significance.
    • Example: iPhone battery life claim comparison.
  • Confidence Intervals: Useful for understanding effect size and statistical significance.
  • Standardizing Effect Sizes:
    • Cohen’s d: Measures effect size in terms of standard deviations.
    • Hedges’ g: Adjustment for small samples (<50); calculates with a correction.
  • Assumptions: Data should come from a normally distributed population.

Errors and Power in Statistical Testing

  • Type I Error: Rejecting a true null hypothesis.
    • Probability denoted by alpha, often set at 0.05.
    • Adjusting alpha affects risk of Type II error.
  • Type II Error: Failing to reject a false null hypothesis.
    • Probability denoted by beta.
  • Statistical Power:
    • Probability of correctly rejecting a null hypothesis (1 - beta).
    • Greater power is desirable but depends on method of achieving it.
  • Distribution Analysis:
    • Null and alternative hypothesis distributions illustrated.
    • Null hypothesis represented by bell curve, significance level (alpha) shaded.
    • Alternative hypothesis represented with hypothetical distribution for illustration.

Upcoming Topics

  • Will continue discussion on statistical power and errors in upcoming class.