Lecture 7PreviewWhat does Jason’s story have to do with statistics?Slide 4To Start…ExampleHow do we decide: More than intuitionHypothesis TestingBasic Logic of Hypothesis TestingUnknown PopulationRules of Hypothesis Testing (1)Slide 12Slide 13Slide 14Slide 15Results of Hypothesis Testing: Uncertainty and ErrorSlide 17Slide 18Slide 19Possible Outcomes of a Statistical DecisionLet’s think about what happens as a result of our decisionLet’s Try OneLet’s Do One: EvidenceIn the LiteratureTry One…Directional Hypothesis TestsHypothesis for Directional TestsCritical Region for Directional TestsTest statistic and Decision: for Directional Hypothesis TestingSlide 30AnswersOne vs. Two tailsStatistical PowerSlide 34Assumptions for the Hypothesis Test with Z-scoresSlide 36Criticisms about Hypothesis TestingSlide 38Slide 39Slide 40Measuring Effect SizeHomework Chapter 8Lecture 7Inferential Statistics: Hypothesis TestingPreviewJason decides to sue the city after he was involved in a near fatal collision at Division St. and Trailridge Dr. Jason is claiming that his accident could have been prevented if the city had placed a stoplight at the intersection. It turns out that this intersection has had an abnormally high number of accidents in the past 5 years. The city is arguing that the number of accidents at this particular intersection is not abnormal and that there are no more accidents at this particular intersection than others in town.What does Jason’s story have to do with statistics?The primary application of inferential stats is to help researchers interpret their data.(1) Are the differences in the data due to chance?(2) Are the differences in the data something more than chance.In the example with Jason(1) Are their an increased number of accidents due to external factors (lack of a stoplight not chance)?(2) Is the number of accidents at that particular intersection no more than chance?Hypothesis testing is a statistical procedure that allows researchers to use sample data to draw inferences about the population of interest–Is the mean of my observed sample consistent with the known population mean or did it come from some other distributionTo Start…Is the mean of myobserved sample consistent with the known population mean or did it come from some other distribution?We are given the following problem:–There exists a sample of cars (some kind)–They get an mean MPG of 19 miles–Are they midsize cars (we can’t go look at them)We know:–A midsize car gets 18 MPG–Is 19 different enough from 18 in this distribution–Or is it part of some other distributionExampleHere’s what we know: = 18–M = 19M = 0.4M - MZ = z = (19 - 18) / 0.4z = +2.5p = .0062 or PR = .62%How do we decide: More than intuitionIf the z-score falls outside the middle of 95% of the curve, it must be from some other distributions (yesterday p<.05 convention in psychology)Main assumption: We assume that weird, unusual, or rare things don’t happenIf a score falls out into the 5% range we conclude that it “must be” from some other distribution. Less than %5 is rare enoughHypothesis TestingHypothesis Testing: a statistical procedure that allows researchers to use sample data to draw inferences about a population–Use the concepts of:•z-scores•probability•distribution of sample meansBasic Logic of Hypothesis Testing(1) State a hypothesis about the population–Hypothesis: prediction about the relationship between variables; how IV affects DV–e.g. People who prefer Hagen Daaz ice cream will have a mean IQ that is higher than average at 130(2) Use the hypothesis to predict the characteristics the sample should have(3) Obtain a random sample– random sampling: when all potential observations in the population have equal chances of being selected.–RANDOM: Survey every 100th house from the list of all addresses in Tucson.–NOT RANDOM: Survey the internet users about access to new technology in their schools(4) Compare sample data with hypothesis–using a statistical test (today we’ll continue to use z-tests, but keep in mind that other statistical stats can be used in a similar fashion)Unknown Population = 24 = ?Known population before treatmentUnknown population after treatmentTreatment = 4 = 4One basic assumption = If the treatment has any effect it is simply to add a constant amount to (or subtract a constant amount from) each individual’s scoreNo change in shape of distribution or standard deviationUnknown pop. is just theoretical (we never administer a treatment to the entire pop.), but we do have a real sample that represents the pop., so this is what we useRules of Hypothesis Testing (1)(1) STATE THE HYPOTHESIS about the unknown population mean.–null hypothesis: H0 = statement that the treatment has no effect; IV has no effect on DV.–alternative hypothesis: H1 = treatment had an effect on DV. Alternative hypothesis does not specify direction of change. It some cases it might be useful to specify (we’ll get to that).–NOTE: the null and alternative hypotheses are mutually exclusive and exhaustive. They can’t both be true.Example: on average the population remembers 7 words in a particular situation with a SD = 2. You test a new intervention designed to enhanced memory with 10 participants and find that they remember an average of 9 words.H0 = new intervention = 7H1 = new intervention = 7Rules of Hypothesis Testing (2)(2) SET THE CRITERIA for a decision–Data will either support or refute the null hypothesis–Distribution gets divided into 2 sections:•Sample means that are likely if H0 is true.•Sample means that are very unlikely if H0 is true –Must set the boundaries that indicate the high-p samples from the low-p samples.•Level of significance or alpha level () make critical region•Convention says  = .05 or 5%, but other commonly used alpha levels are .01 (1%) and .001 (0.1%).•A z-score can mark the boundary set by alpha!* Because the extreme 5% can be split between 2 tails there is 2.5% or .025 in each tail (2-tailed)(3) COLLECT DATA and compute sample statistics–Select a random sample from the population–NOTE: it is important to collect the data after stating the hypothesis and establishing the criteria in order to make an objective evaluation of the data–Compute a sample/ test statistic (today we are illustrating hypothesis testing through a sample statistic