Lesson 10: T-Tests

Aug 30, 2013 ../st095-statistics

Lesson 10: T-Tests

First half

  • Basic premise for t-test

    1. Get the sample mean to compare to the population mean (or alternate sample)
    2. Find the sample standard deviation
    3. Use it to calculate t value

    t = (mu - sample_mu) / (std_dev / sqrt(n))

    1. Look up t-table to find critical p-value for your alpha level.
      • Degrees of freedom = sample_size - 1
      • If it's a two-tailed test, then: alpha level / 2
    2. Is the t score further away from 0 than the critical probability?
    3. If so, then it's statistically significant. Or, we reject the null hypothesis
    4. Determine the sample standard deviation using Bessel's Correction
    5. S = sqrt( variance / (n - 1) )
    6. t-distribution
    7. more prone to error
    8. more spreadout
    9. the larger n is (the sample size)
      • the closer the t-dist is to normal
      • the tails get skinnier
      • less margin of error
    10. Understanding degrees of freedom
    11. Example: if you have 3 marbles to put in 3 cups
      • 1st cup: 3 choices of marbles
      • 2nd cup: 2 choices of marbles
      • 3rd cup: 1 choice
      • Therefore, the last cup is forced, so you have 2 degrees of freedom
    12. Finch example (birds)
    13. Scientists map a trait of the birds like beak width
    14. Average beak width = 6.07mm
    15. Do Finches today have different-sized beak widths than before?
    16. Null = beak width == 6.07mm
    17. Alternate = beak width != 6.08mm
    18. Sample size = 500, df = 499
    19. x-bar = average_of_sample = 6.4696
    20. Std dev = sqrt(variance(sample)) = 0.4
    21. t-statistic = (x-bar - mu) / (Std_dev / sqrt(n)) = 22.36
    22. We can definitely reject the null

    * Cohen's d * Common measure of "effect" size when comparing means * Measures the distance between two means in std deviation units * Instead of dividing by standard error, divide by standard deviation of the sample * Dependent samples * "When the same subject takes the test twice" * Two different treatments * Pre-test, post-test * Growth over time (longitudinal study)

New half

  • Effect Size
    • size of treatment effect
      • if you have a treatment variable, what's the difference between two means?
    • everyday meaning
      • variables you can understand without special training
    • types of effect size measures
      • difference measures
      • standardized differences
        • Cohen's d
      • correlation measures
        • r2
          • "proportion (%) of variation in one variable that is related to ('explained by') another variable"
  • Statistical significance
    • Rejected the null
    • Results not likely due to chance (sampling error)
  • Cohen's d
    • Provides "standardized mean difference"
    • d = (x-bar - Mu) / std
    • Interpretation: how far apart the sampling mean is in standard deviations

  • R-squared - r^2 - coefficient of determination

    • Result: 0.0 - 1.00
      • 0 == variables that are not related
      • 1 == variables that are perfectly related (near impossible)
    • r^2 = t^2 / (t^2 + df) Note: t-score is not t-critical value
      • Example:
      • ```t = 2, df = 24 == 4 / (24 + 4) == 0.167

  • Results section