Determining the Independence of Variables & Variates Much quantitative research compares distributions (empirical/ theoretical) But are the variables/ variates independent? Importance: the kind of hypothesis testing & assessment of probability related to independence Comparing: Surface area of skeletal elements collected together at a point along a river Heart rates of individuals before/ after each climbs four flights of stairs Opinions about a bill to tax farm effluents among adult males in a small farming community

Determining the Independence of Variables & Variates Key question to ask: How reasonable is the assumption of independence in light of the null hypothesis? Subsistence strategy and post-marital residence among societies that all share a “recent” common origin? Opinions about an issue among people who have never associated Change over a five year period in the percentage of people involved in wage labour in several communities in Mexico.

Determining the Independence of Variables & Variates Steps to take to figure this out: Think about what you need to know? Become familiar with related literature surrounding an issue. State your variable definitions and assumptions in such a way that both you and others can evaluate them Plan your choice of statistical tests around expectations about independence or its absence* * Yes, more on these tests later

Problem Statement You now have: Determining relevant problem in the literature Articulating a clear hypothesis Considering variables, operational definition & independence Statistical population, samples, and sampling methods

Probability Descriptive & inferential stats depend on probability And probability based on assumption of empirical/ theoretical regularities in world

Theoretical Determination of Probability Coin example: With a fair coin flip, 50% chance heads, 50% tails P(H)=0.5, P(T)=0.5 with summed probabilities= 1 Dice example: With fair dice, total of 36 combinations Only one way to role a two, 1/36 = 0.03; two ways to role a three, 2/36 = 0.05, etc. Determine all logical possibilities

Empirical Determination of Probability Calculate probabilities by measuring actual trials A concern: are you characterizing a regularity that applies to relevant space and time?

Complex Events & Determination of Probability Deck of card example: Individual card, 1/52, e.g., P(2♣)=0.02 P(♣)=13/52=0.25 Complex Events (more than one) : P(♣Ŭ♠)= 13/52+13/52 = 0.50 Complex Events (consecutive independent successes) P(n events) = (event freq. / trials)n P(♠twice in row)= (13/52)x(13/52)=0.06

The Binomial Distribution & Empirical Probabilities Binomial distribution formalizes calculation of probabilities when two possible outcomes, and specific number of trials P(p+q)n For 3 trials: (p+q)x(p+q)x(p+q) p3 + 3p2q + 3pq2 + q3 Randomly grading the labs, what is the probability that you will be the top scorer at least once over the first three labs? 1 of 24 students p = 0.04, q = 0.96 Plug in p and q for terms than include at least one success = 0.12

Probabilities for Events & Probabilities for Distributions Calculate probability of events under assumptions of regularity and consistency (theoretically or empirically determined) Determine how likely our event is, and then interpret it accordingly. However, often want to know how likely is a distribution given assumptions of regularity? Or, how likely is this variate in a distribution of variates? Answering these kinds of questions require a theoretical foundation in the normal distribution.