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

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 = 187.2 mm, s = 7.9, n = 12 Newly found skull: length 175 mm The t-score for comparing variate and sample mean: t = 175-187.2/7.9 t = -1.54 (does this exceed critical value?)

Hypothesis Testing with the t-Distribution: example > qt(.975, 11) [1] 2.200985

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) = 204.6 & 169.8 Yskull_i 169.8 204.6

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

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)