1. Hypothesis Testing with the t-Distribution
t-distribution Review
All previous hypothesis tests assume we know population parameters
If we don’t know a parameter like µ we can estimate using Confidence Limits and measure of dispersion, σȲ
L = µ ± (1.96 x σȲ ) (for 95% CL)
And we can calculate σȲ by σ/√n
If we don’t know σ we estimate with sample statistic like s.
When we do this, no longer have single distribution with which to calculate area under the curve
t-distribution changes with degrees of freedom

2. Hypothesis Testing with the t-Distribution: example
Ho: Yi = Ȳ
Common in bioanthropology and archaeology
Use critical value of t to test
Ho: Is length of skull indistinguishable from all burials at cemetery?
Easter Island cemetery skull mean length, Ȳ = mm, s = 7.9, n = 12
Newly found skull: length 175 mm
The t-score for comparing variate and sample mean:
t = /7.9
t = (does this exceed critical value?)

3. Hypothesis Testing with the t-Distribution: example
> qt(.975, 11)
[1]

4. Hypothesis Testing with the t-Distribution: example
Fail to reject H0: Yskull_i = ȲEIskulls
Can also calculate Confidence Limits around the sample mean (Ȳ) and see if Yi is within them
187.2 ± (2.201 x 7.9) = &
Yskull_i
169.8
204.6

5. Hypothesis Testing with the t-Distribution
H0: Yi = Ȳ
H0: Ȳ = µ
And we do not know σ
Variation considered is between variate and Ȳ: this is quantified using s
Variation considered is between Ȳ and µ: this is quantified using sȲ

6. One-Tailed Null Hypotheses
All of our hypotheses have been two-tailed
Is a variate/mean indistinguishable from another variate/mean?
We split the rejection region into each tail of distribution
One tailed hypotheses put all of rejection region in one tail
H0: Ȳ ≤ µ & the “hypothesis of interest”
Tables of critical values assume two-tailed test
For one-tailed, must double alpha when looking up critical value (except in R)