PSY 307 – Statistics for the Behavioral SciencesWhy Hypothesis Tests?Kinds of Hypothesis TestingHypotheses are about Underlying PopulationsGuessing about the PopulationCommon vs Rare OutcomesCommon vs RareHypothesis TestingManipulationsNull HypothesisAlternative HypothesisBoundaries for OutcomesDifferent Critical Values can be UsedCritical Values can be StricterOr More LenientDecision RuleDecision and InterpretationOne-Tailed & Two-Tailed TestsOne-tailed vs Two-tailed TestsOne-Tailed Test (Upper Tail Critical)One-Tailed Test (Lower Tail Critical)Two-tailed TestOne-tailed TestOne-tailed Test (other direction)Slide 25Hypothesis Test OutcomesPossible DecisionsStrong and Weak DecisionsDecisions are Usually CorrectProbabilities of ErrorPower CurvesDeciding Which to UsePSY 307 – Statistics for the Behavioral SciencesChapter 10-11 – Introduction to Hypothesis TestingWhy Hypothesis Tests?The first step in any study is to test against chance.We cannot draw any conclusions about our results without making sure our results are not accidental.We never know for sure what the true situation is, but we try to minimize possibility of error.Kinds of Hypothesis TestingTesting against 0 (zero)Testing against chanceTesting the null hypothesis versus an alternative hypothesisTesting two hypotheses that predict different or incompatible outcomesModelingExploratory data analysisHypotheses are about Underlying PopulationsUnderlying PopulationSampleWhat can we know about the population by measuring the sample?Guessing about the Population? ?Null PopulationAlternative Population(s)SampleWhich population is the true underlying population for our sample? We assume the null population is true, then see how likely the sample would be to occur under those circumstances.We cannot know for sure.Common vs Rare OutcomesWe always start with the assumption that the null population is true. In that case we ask:How likely would our current result be if the null population were the true underlying population?A result in the center of the normal curve is highly likely (common).A result in the tails of the normal curve is much less likely (rare).Common vs Rare-1.96 1.96RARERARECOMMONWe use +/- 1.96 (2 SD) as the cutoff between common and rare.All possible sample resultsHypothesis TestingA hypothesis is a prediction about the results of a study.Null hypothesis (H0) – a prediction that the null population is the true underlying population.The null is a hypothetical sampling distribution – what would exist if nothing special were happening.Results are compared against it.ManipulationsA comparison is being made:A sample against a known population.Two groups are compared with each other (e.g., males vs females).Two groups are compared after a manipulation (e.g., treatment vs control.The expected differences are stated as the alternative hypothesis (H1).Null Hypothesis-1.96 1.96RARERARECOMMONNull hypothesis is true (always assumes no difference)Alternative Hypothesis-1.96 1.96RARERARECOMMONNull hypothesis is NOT true (a difference most likely exists)Boundaries for OutcomesCommon outcome – small difference from the hypothesized population mean .Rare outcome – too large a difference from the hypothesized mean to be probable.A set of boundaries (critical values) can be found to decide whether an outcome is rare or common.Different Critical Values can be Used-1.96 1.96RARERARECOMMONNull hypothesis is NOT true (a difference most likely exists)p < .05Critical Values can be Stricter-2.58 2.58RARERARECOMMONNull hypothesis is NOT true (a difference most likely exists)p < .01Or More Lenient-1.65 1.65RARERARECOMMONNull hypothesis is NOT true (a difference most likely exists)p < .10Decision RuleThe decision rule specifies precisely when the null hypothesis can be rejected (assumed to be untrue).Critical score – the boundary for a rare outcome.Most studies in psychology use a critical value of p < .05 unless they have a good reason not to.Level of significance () – rarity.Decision and InterpretationReject the null hypothesis when the observed mean results in a test statistic beyond the critical value.If your level of significance is p<.05 then a z-score above 1.96 or below -1.96 is rare.With a rare result, conclude that the null hypothesis is untrue (reject it).With a common result, retain it.One-Tailed & Two-Tailed TestsTwo-tailed tests (non-directional) divide the probability of error () between the two tails.Expressed using equality signs (=, /=).One-tailed tests (directional) place all of the probability of error in a single tail in the direction of interest.Expressed using inequality signs (<, >=)One-tailed vs Two-tailed TestsA two-tailed test only predicts a difference:H0:  = 25, 1 = 2H1:  ≠= 25, 1 ≠ 2A one-tailed test predicts a difference in a specific direction:H0:  ≤ 25, 1 ≤ 2H1:  > 25, 1 > 2One-Tailed Test (Upper Tail Critical) 1.65RARECOMMONNull hypothesis is NOT true (a difference most likely exists)p < .05One-Tailed Test (Lower Tail Critical)-1.65RARECOMMONNull hypothesis is NOT true (a difference most likely exists)p < .05Two-tailed Test-1.96 1.96.025.025COMMON = .05One-tailed Test 1.65.05COMMON = .05One-tailed Test (other direction) -1.65.05COMMON = .05Two-tailed Test-1.96 1.96.025.025COMMON = .05Hypothesis Test OutcomesA hypothesis test has four possible outcomes:H0 is true and H0 is retained – a correct decision.H0 is true but H0 is rejected – a Type I error (false alarm).H1 is true and H0 is rejected (H1 is retained) – a correct decision.H1 is true but H1 is rejected and H0 is retained – a Type II error (miss).Possible DecisionsReality – Null is TrueReality –Null is FalseWe decide that the Null Hypothesis is TrueCorrectly retain nullRetain null but make aType II error (miss)We decide that the Null Hypothesis is FalseReject null and make aType I error (false alarm)Correctly reject nullStrong and Weak DecisionsRetaining the null hypothesis (H0) is a weak decision because it is ambiguous and uninformative.H0 could be true or there may be a difference but the study couldn’t demonstrate it, due to poor methods.Rejecting the null hypothesis is a strong decision because it implies that H0 is probably false.Decisions are Usually CorrectWe never actually know what is