PSY 307 – Statistics for the Behavioral SciencesDemosA Family of Normal CurvesDifferent Normal CurvesZ-ScoreProperties of Z-ScoresStandard Normal CurveFinding ProportionsFinding Exact ProportionsOther DistributionsWhy Samples?A Sample comes from a PopulationRandom SamplesTechniques for Random SelectionHypothetical PopulationsRandom AssignmentHow to Assign SubjectsProbabilityAddition RuleAddition Rule (Cont.)Multiplication RuleDependent OutcomesProbability and StatisticsPSY 307 – Statistics for the Behavioral SciencesChapter 8 – The Normal Curve, Sample vs Population, and ProbabilityDemosHow normal distributions are generated:http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.htmlHow changes in the mean and std deviation affect the shape of the normal distribution:http://onlinestatbook.com/chapter6/varieties_demo.htmlFinding the proportion for a given z score: http://onlinestatbook.com/java/normal.htmlFinding the z-score for a given portion of the distribution:http://onlinestatbook.com/java/normalshade.htmlA Family of Normal CurvesA normal curve has a symmetrical, bell-like shape.The lower half (below the mean) is the mirror image of the upper half.Values for the mean, median and mode are always the same number.The mean and SD specify the location and shape (steepness) of the normal curve.Different Normal CurvesSame SD but different Means Same Mean but different SDsZ-ScoreIndicates how many SDs an observation is above or below the mean of the normal distribution.Formula for converting any score to a z-score:Z = X –  Properties of Z-ScoresA z-score expresses a specific value in terms of the standard deviation of the distribution it is drawn from.The z-score no longer has units of measure (lbs, inches).Z-scores can be negative or positive, indicating whether the score is above or below the mean.Standard Normal CurveBy definition has a mean of 0 and an SD of 1.Standard normal table gives proportions for z-scores using the standard normal curve.Proportions on either side of the mean equal .50 (50%) and both sides add up to 1.00 (100%).Finding ProportionsActually +/-1.96Finding Exact Proportionshttp://davidmlane.com/hyperstat/z_table.htmlhttp://www.sfu.ca/personal/archives/richards/Table/Pages/Table1.htmOther DistributionsAny distribution can be converted to z-scores, giving it a mean of 0 and a standard deviation of 1.The distribution keeps its original shape, even though the scores are now z-scores.A skewed distribution stays skewed.The standard normal table cannot be used to find its proportions.Why Samples?Population – any complete set of observations or potential observations.Sample – any subset of observations from a population.Usually of small size relative to a population.Optimal size depends on variability and amount of error acceptable.A Sample comes from a PopulationRandom SamplesTo be random, all observations must have an equal chance of being included in the sample.The selection process must guarantee this.Random selection must occur at each stage of sampling.Casual or haphazard is not the same as “random.”Techniques for Random SelectionFishbowl method – all observations represented on slips of paper drawn from a fishbowl.Depends on thoroughness of stirring.Random number tables – enter the table at a random point then read in a consistent direction.Random digit dialing during polling.Hypothetical PopulationsCannot be truly randomly sampled because all observations are not available for sampling.Treated as real populations and sampled using random procedures.Inferential statistics are applied to samples from hypothetical populations as if they were random samples.Random AssignmentRandom assignment ensures that, except for random differences, groups are similar.When a variable cannot be controlled, random assignment distributes its effect across groups.Any remaining difference can be attributed to effect, not uncontrolled variables.How to Assign SubjectsFlip a coin.Choose even/odd numbers from a random number table.Assign equal numbers of subjects to each group by pairs:When one subject goes to one group, the next goes to the other group.Extend the same process to larger numbers of groups.ProbabilityThe proportion or fraction of times a particular outcome is likely to occur.Probabilities permit speculation based on observations.Relative frequency of heights also suggests the likelihood of a particular height occurring.Probabilities of simple outcomes are combined to find complex outcomesAddition RuleUsed to predict combinations of events.Mutually exclusive events are events that cannot happen together.Add the separate probabilities to find out the probability of any one of the outcomes occurring.Pr(A or B) = Pr(A) + Pr(B)Addition Rule (Cont.)When events can occur together, addition must be adjusted for the overlap between outcomes.Add the probabilities then subtract the amount that is shared (counted twice):Drunk drivers = .40Drivers on drugs = .20Both = .12Multiplication RuleUsed to calculate joint probabilities – events that both occur at the same time.Birthday coincidencehttp://www.cut-the-knot.org/do_you_know/coincidence.shtmlPr(A and B) = [Pr(A)][Pr(B)]The events combined must be independent of each other.One event does not influence the other.Dependent OutcomesDependent – when one outcome influences the likelihood of the other outcome.The probability of the dependent outcome is adjusted to reflect its dependency on the first outcome.The resulting probability is called a conditional probability.Drunk drivers & drug takers example.Probability and StatisticsProbability tells us whether an outcome is common (likely) or rare (unlikely).The proportions of cases under the normal curve (p) can be thought of as probabilities of occurrence for each value.Values in the tails of the curve are very rare (uncommon or