PSYC200; Exam 2 GuidePopulation Distributions In a sample: observed (frequency) In a population: theoretical (probability)- Continuous Distributions – describe an infinite number of possible data valueso Continuous curves- Discrete Distributions – describe a finite number of possible values o HistogramsRandom Sampling Methods – gives the most unbiased samples; parts of population that are over-represented are by chance- Simple random sample – every observation has an equal chance of being sampled; sampled independently (w/ replacement)- Systematic random sample – randomly select a starting point and continue selection by a “rule” o ex: every 3rd person in a sample- Stratified random sample – the total population is divided into groups (strata) based on special interests (race, gender) samples are taken from each groupo Used for variables that are difficult to compareo Strata are usually deliberately over-represented- Cluster sample – similar to stratified, but division of groups occurs in a convenient way (physical location) Subjective Probability- personal view/belief of what we think the probability isObjective Probability- actuality, what the probability actually isSimple Probability – the likelihood of some event happening compared to all possible events happening*Probability is a measureVenn Diagrams- Union (“or”) – refers to one or the other event- Mutually exclusive – 2 events have no outcomes in commono This intersection in null; p(A and B) = 0- Intersections (“and”)- when events have outcomes in common- Exhaustive – a set of events that includes all possible outcomes Addition Rule: p(A OR B) = p(A) + p(B) – p(A and B)- If events are mutually exclusive, the addition rule is reduced to: p(A or B)= p(A) + p(B)Multiplication Rule: p(A AND B) = p(A) * p(B | A)- Joint probabilities –likelihood of observing each of the 2 events- Conditional probabilities (“given”) – the likelihood that an event will occur given that some other event occurs OR AdditionAND Multiplication*Equality = independence *Inequality = dependence p(A|B) ≠ p(B|A) UNLESS p(A) = p(B)Independent Events – if occurrence of one event has no effect on the probability of occurrence of other event- Events are independent if and only if: o p(A and B) = p(A) * p(B) p(B) = p(B|A)- If events A and B are independent, then multiplication rule is reduced to: o p(A AND B) = p(A)*p(B)- For sampling without replacement= non-independent events- For sampling with replacement= independent events Bayes’ Rule Probability of desired event over all possibilities Used in diagnosis, medicine, public policy, etc Probabilities based on relative frequency dataProbability DistributionsNormal Distributions Parameters:o μ= population meano σ^2= population variance Fully described by its mean and standard deviation Shape describes many existing variables (i.e. weight) For continuous distribution Symmetric about the mean o Approx. 68.3% of the observations are within ± 1 s.d. of the meano Approx. 95% of the observations are within ± 2 s.d. of the mean Area under the curve = 1Table A gives the areas of all z scores (z-scores transform raw scores into the # of s.d. away from the mean) Column A: z-scores (z = x – mean / standard deviation) Column B: area between mean and z-score Column C: area above the z-score (z-score to tail) p(X or higher) get (+) z-score “C value” p(X or higher) get (-) z-score “B value + 0.5” p(X or lower) get (+) z-score “B value + 0.5” p(X or lower) get (-) z-score “C value” p(between X and Y) get the z scores of X and Y add both B values together p(not between X and Y) get the z scores add both C values togetherBinomial Distributions When you count how many of a sample of a fixed size have a certain characteristic (i.e. 3 chances to pullout 3 twix); WITH replacement For discrete distributions N = fixed sample sizeP = successQ = failure Requirements:o Series of N trialso Each trial has only 2 possible outcomeso On each trial the two outcomes are mutually exclusiveo There is independence between the outcomes of each trialo The probability of each outcome remains constant from trial to trial Mean = np Variance = npq Standard Deviation = √npq*The binomial approaches the normal distribution when p = 0.5, as N (# of trials) gets larger*Use normal approximation when np ≥ 10 AND nq ≥ 10Permutations- how many ways can I get X in a certain # of trials?Discrete Distributions (other than the binomial) Poisson – # of successes (x) is random, p is fixed, n is fixed (usually a time period), failures aren’t measured (i.e. Have 30 seconds to pull out some snickers); WITH replacement Pascal – # of trials is random (how many trials must I do to get X successes?), p is fixed, # of successes is fixed (i.e. how many trials to get 3 twix from the bag)o Mean= x/po Variance= x(1-p)/p^2o Geometric- only looking for one success (how many trials until I get my 1st success?) Multinomial – when you have multiple categories/nominal measurement, p is fixed, WITH replacement(i.e. pull out candy, write it down, replace each time) Hypogeometric – equivalent to the multinomial, except p (probability of success) is not fixed, WITHOUT replacementContinuous Distributions Normal compare to population mean T-Distributions compare to another sample both are questions about means F-Distributions compare to another sample Chi-Square compare to variance of population both are questions about variance Hypothesis Testing- procedure that uses sample data to evaluate a hypothesis about a population5 Steps1. State the hypothesis H0 null hypothesis; always states that there is no treatment effect, change, etc. o µ = np H1 alternative hypothesis; states that there was a treatment effect. o µ≠np, µ > np, or µ < np Hypothesis always in terms of population2. Set decision criterion Alpha = level of error you are willing to accept= 0.05 One tailed if directional (< or >, will usually be >) Two tailed if nondirectional (≠)3. Obtain Sample Data Use descriptive statistics4. Calculate the Test Statistic Gives probability value Depends on what the distribution is5. Decide to “reject” or “fail to reject” the null & interpret If test statistic
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