Tues Nov 3 - Statistics Lecture Notes

• Go over homework; answer questions
• Discuss mid-term project assignment
• This week: Student's t-Test
  • Several variations; more complicated formulas.
  • This is the "real thing" - one of the more common hypothesis testing tools.
  • New concept: "degrees of freedom" : how many independent variables
    one sample: N - 1
    unpaired two sample: N1 + N2 - 2
    paired two sample (subtract corresponding numbers) : N - 1
    
  • Basic idea is that with small N, our estimate of the standard deviation isn't very good - sigma is also "fuzzy". This takes that into account, depending on N.
  • Critical values of t are given in Table D-5 in textbook
    ... but in practice you'd usually let a statistics tool do the work for you.
  • Variations described in the text :
    * one sample mean (least common)
    * unpaired two sample assuming equal variances
      (Allison Lennox, one of Jenny's student's asked how to do this just this semester)
    * paired two sample
• Aside : "p-value"; see pg 235 in text.
  • Minor variation on our procedure, slightly different but essentially same.
  • Instead of critical values, find probability (one or two tail) of a value at least as extreme as the one you found. This probability is the p-value.
  • Then accept or reject by comparing p directly to alpha.

• Complications for t-test.
  • we typically use a "pooled estimate" of the standard deviation
  • if "before", "after" data, better to use a "paired" test and look at differences in corresponding values.

• An example done 3 ways: {10, 12, 12, 13} and {9, 10, 12, 12}, two tail means tests, H0 is that these are the same.
  1. old "normal" method. (But with N this small, this isn't quite right.)
       m1=11.75,s1=1.26,m2=10.75,s2=1.5, diff=1.0, sigma_diff=0.98,
       p-value=2*(1-NORMDIST(1.0,0,0.98,TRUE))=0.33
       The NORMDIST gives area to the left of x=1.0 for mean=0, sigma=0.98 normal curve.
       (1-NORMDIST) is the area to the right. 2*(1-NORMDIST) is the two-tailed extreme probability.
  2. unpaired t-test (assuming nothing about list order)
       result: t=1.02, sigma_pooled=1.38, 6 degrees of freedom, p-value=0.346
  3. paired t-test (assuming elements of lists match)
       result: t=2.45, 3 degrees of freedom, p-value=0.092
  4. There's an excel version of these results in the t-test/ directory. Or you can type numbers into the website give below for the t-test results.

• Student's t-TestCalculation tools and references
  • See http://www.physics.csbsju.edu/stats/
    for an online tool to do two-tailed paired/unpaired tests.
    (That site also has some good explanations.)
  • Also see "Data Analysis" menu in Excel.
    * t-Test: Paired two sample for means
    * t-Test: Two-sample assuming equal variances
    * t-Test: Two-sample assuming different variances (rarer; not in text)