- 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.
- 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. - unpaired t-test (assuming nothing about list order)

result: t=1.02, sigma_pooled=1.38, 6 degrees of freedom, p-value=0.346 - paired t-test (assuming elements of lists match)

result: t=2.45, 3 degrees of freedom, p-value=0.092 - 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.

- old "normal" method. (But with N this small, this isn't quite right.)
**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)

- See