Making Inferences From z to t Statistics

A Brief Review of Significance Testing Revisit Alpha (Type I error) Type II error 1 – b = Power (Power)

Degrees of Freedom X S = 80 10 10 10 10 10 10 10

Selecting the Proper Test Level of data? Descriptives vs. Inferentials vs. Nonparemetrics Comparison? Correlation? Prediction? Regression vs. Pearson’s R vs. Anova or t How many groups, variables, or levels? Anova vs t Design? Covariates Inflated Type I error? Multivariate vs Univariate tests

Student’s t Test Against Population Means Easy to calculate When you know the mean and standard deviation of a sample, and wish to compare them to a known population mean t = (M – m)/s df = n – 1 If tobt ≤ tcrit, then significant at alpha

A Real-Live Example Tomak et al. (2009) compared MMPIs of a sample of Internet sex offenders (n =48) versus those of a previously studied comparison group of general sex offenders (Summerhill, 2002; unpublished dissertation). In hindsight, needed to determine if groups differed on demographic variables, such as age. Did not have Summerhill’s raw data, but did have the mean: 45.96 Andjelkovic’s demographics gave us M = 40.67, s = 11.37 Is this a significant difference? Nope: t(47) = -0.465, p > 0.05

The Independent t Test The t vs. population means are convenient, but not terribly powerful. If you have the raw data of independent groups, this formula is much better:

The Paired t Test Similar philosophy as the independent t test, but used when subjects in one group are matched, yoked, used in repeated measures designs, or are otherwise “connected” to subjects in another group. df = n - 1

X1 55 43 51 62 35 48 58 45 54 56 32 X2 48 38 53 58 36 42 55 40 49 50 25 D SD = D2 SD2 = 7 49 5 25 -2 4 4 16 1 -1 6 36 3 9 5 25 -1 1 4 16 -2 4 7 49 35 235

Now, The Fun Part SD/√[(N(SD2) – (SD)2)/n – 1] 35/√[(12(235) – (35)2)/12-1] 35/√[(2820 – 1225)/11] 35/√1595/11 35/√145 35/12.042 2.906 df = 12 – 1 = 11 tcrit = 2.201 (a = 0.05, 2-tailed) Reject the null

Assumptions… Normality Linearity Interval-level data

A Cup of t Test Determine the best t test to use on the provided data set and interpret the results.

t test against population means Hotelling’s T You have RBANS scores of an elder population that you wish to compare to established norms. Which assessment would be best? Paired-samples t test Independent t test t test against population means Hotelling’s T t test against population means

What advantage does the t test have over the Analysis of Variance? Ease of calculation Greater power Lower error rate Ability to compare related groups Ease of calculation

Questions? Thoughts?