Introduction to Hypothesis Testing: The Binomial Test 9/29

ESP Your friend claims he can predict the future You flip a coin 5 times, and he’s right on 4 Is your friend psychic?

Two Hypotheses Hypothesis Psychic Luck Hypothesis testing A theory about how the world works Proposed as an explanation for data Posed as statement about population parameters Psychic Some ability to predict future Not perfect, but better than chance Luck Random chance Right half the time, wrong half the time Hypothesis testing A method that uses inferential statistics to decide which of two hypotheses the data support

Likelihood Likelihood Determining likelihood Strategy: Probability distribution of a statistic, according to each hypothesis If result is likely according to a hypothesis, we say data “support” or “are consistent with” the hypothesis Determining likelihood Psychic: hard to say; how psychic? Luck: can work out exactly; 50/50 chance each time Strategy: Work out probabilities according to luck hypothesis Calculate how likely outcome would be Likely: luck can explain it Unlikely: luck can't explain; must be ESP “Innocent until proven guilty” Assume luck unless compelling evidence to rule it out Population Sample Probability Inference Hypotheses Statistics Likelihood Support

Likelihood According to Luck 6/32 = 18.75% Flip 1 2 3 4 5 Y N

Binomial Distribution Binary data A set of two-choice outcomes, e.g. yes/no, right/wrong Binomial variable A statistic for binary samples Counts how many are “yes” / “right” / etc. Binomial distribution Probability distribution for a binomial variable Gives probability for each possible value, from 0 to n A family of distributions Like Normal (need to specify mean and SD) n: number of observations (sample size) q: probability correct each time

Binomial Distribution Formula (optional): n = 20, q = .5 n = 5, q = .5 n = 20, q = .5 n = 20, q = .25

Binomial Test Hypothesis testing for binomial statistics Null hypothesis q = .5 Nothing interesting going on; blind chance Alternative hypothesis q equals something else Find likelihood according to null hypothesis p(count  c) Special number, called p-value or p Tells how good an explanation for the data the null hypothesis is If p > .05 Null hypothesis can explain data If p < .05 Null hypothesis cannot explain data Rule out null and believe alternative hypothesis .05 is arbitrary cut-off; more on this next time

Applications Any question of rate or proportion Population Parameter Rates of outcomes: coins, other probabilities Answers relative to chance: learning, memory, etc. Composition of group: sex, politics, attitudes Population All people, events, etc. Often infinite (probabilities) Parameter Proportion or probability in population Sample Set of binary outcomes Answers for one student; category for each subject Statistic Proportion or count in sample

Testing for ESP Null hypothesis: Luck, q = .5 Alternative hypothesis: q > .5 Likelihood according to null: Binomial(5, .5) p > .05 Retain null hypothesis Luck can explain the data .15625 .03125

Testing for ESP – Stronger Evidence Nine correct out of ten Null hypothesis: Luck, q = .5 Alternative hypothesis: q > .5 Likelihood according to null: Binomial(10, .5) p < .05 Luck can’t explain the data Reject null hypothesis .010 .001

Other values of q Multiple-choice tests; guessing card suits New null hypothesis, likelihood function n = 20, q = .25 p = .014

Normal Approximation For large n, Binomial is nearly Normal Consequence of Central Limit Theorem: count = mean  n Mean = nq, Standard Deviation = Convert cutoff to z-score . Use Standard Normal to find p Cutoff = 8.5 p count n q Mean SD cutoff z Approx Exact 4 5 .5 2.5 1.1 3.5 .89 .186 .188 9 10 5.0 1.6 8.5 2.21 .013 .011