1 The t-distribution

General comment on z and t

Moving from z to t  Same concept, different assumptions  Can only use z-tests if you know population SD  So when you have to estimate σ, use t-dist.  t-test estimates population SD from sample SD  t-test more robust against departures from normality (doesn’t affect the accuracy of the p-estimate as much) 1 2

Calculating the t-statistic  We don’t know the population SD ? Step 1: Estimate σ with “s”

Calculating the t-statistic  We don’t know the population SD ? Step 2: Use “s” to estimate SE M 1 2 3

Calculating the t-statistic  We know the population mean, but not the SD… ? Step 3: Use in t-statistic 1 2

t-statistic – factors in significance  Size of estimated SE obviously depends on both SD of sample, and sample size  Thus, factors affecting size of calculated t are mean diff, sample SD, and sample size 1 2 3

Sampling distribution of t  Moving from the z-distribution to the t- distribution  Still about estimating probabilities, but the properties of the z- and t- distributions are different 1 2

Sampling distribution of t  The t distribution varies with sample size  The good old 1.96 for 95% is toast 1 2

Sampling distribution of t df = n-1 (see next 2 slides) Because distribution gets flatter as n gets smaller, this implies t for significance gets bigger as n gets smaller α (Significance level)

(digression – degrees of freedom)  Degrees of freedom  The number of independent pieces of information a sample of observations can provide for purposes of statistical inference  Why doesn’t d of f = “n” (sample size)?  Not all “n” free to vary  You “give up” one degree of freedom for every population parameter you use the sample to estimate

(digression – degrees of freedom)  Degrees of freedom  Got it? No?  Look, it’s really about high dimensional geometry anyway, so just be content with this reality:  Why does this matter?  DF are used to give an estimate of the accuracy of making a prediction, from your sample, of the population…the more DF you have, the more accurate this prediction will be (and therefore the more likely it will be that you get significant results) 1 2 3

Sampling distribution of t 1

Overall logic again  Get your critical t from the table  Calculate your actual t using  If the calculated t is more different from 0 than the table t, you reject the null