PSYC 612, SPRING 2010Lecture 6: Hypothesis TestingLecture Date: 10/ 06/2010Contents1 Preliminary Questions 12 Part I: Hypothesis Testing Review (75 minutes; 10 minute break) 22.1 Distribution of sample means and the sampling distribution . . . . . . . . . . . . . 22.2 Describe the distribution of sample means . . . . . . . . . . . . . . . . . . . . . . . 22.3 Sample means have z-scores as well . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Z-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 State the hypotheses and define the critical region . . . . . . . . . . . . . . . . . . . 42.6 Conduct a hypothesis and make a statistical decision . . . . . . . . . . . . . . . . . 42.7 Type I and Type II errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.8 Directional hypothesis tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.9 Effect sizes and statistical power (time permitting) . . . . . . . . . . . . . . . . . . 62.9.1 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9.2 Statistical power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Part II: Advanced Material not in the readings (25 minutes; 5 minute break) 74 Part III: Advanced Optional Material (30 minutes) 81 Preliminary Questions•Have you read all the assigned reading f or today?•Have you scheduled your module?•Did you complete the Q&A form?12 Part I: Hypothesis Testing Revi ew (75 minutes ; 10 minutebreak)2.1 Distribution of sample means and the sampling distributionThe key terms you ought to know by now are:• sample• population• random sample• mean• standard deviation2.2 Describe the distribution of sample means• distribution of sample means - a curve that represents the means of samples• sampling distribution - the shape of the curve that comes from selecting all possible samples(fixed size) fro m a population• central limit theorem - as Nsamples→ ∞,¯¯x = µ for any population no matter what shape, mean or standard deviation the distribution becomes normal rat her quickly (Nsamples≥ 30) and will always be normalif 1) the population sampled is no r ma l and 2) the number of samples is relatively large(again, greater than or equal to 30)• expected value of barX (usually noted as E(¯x)) - the mean of the distribution of sample meansor the mean of means (¯¯x)• standard error of the mean ( SE¯xor σ¯x) is the difference between the mean of means (¯¯x) andthe population mean (µ)• the law of large numbers - as the sample size increases, the more likely the sample mean will beclose t o the population meanThe last point is relevant to our next majo r statistic - the standard error of the mean. If thesample size dictates the distance between the sample mean and the population mean then we knowone thing. As nsampleapproaches infinity, ¯x = µ.We have a simple formula that helps us relate how far our mean is from the population meanby using the standard error of the mean:σ¯x=s√nNote we use n here and no t df = n − 1. This is very important. We just want to tie theseconcepts back to the law of larg e numbers and the centra l limit theorem.2.3 Sample means have z-scores as wellSince the sampling distribution has a mean and a standard deviation, we can standardize any samplemean based upon the expected population parameters. The z-score remains the same formula aswe had before except we now use:2z −score =¯x − µσ¯xWhat does this do for us? It allows us to cha r acterize how far away our sample mean is fromthe expected population mean and we can use a time-trusted z table (see Appendix A B in RR).2.4 Z-testNow let us do a simple problem to make sure we all can follow the logic of hypothesis testing.Consider a random sample of 4 scores obtained from a normal population with a mean of 20 anda standard deviation of 4. What is the probability of obtaining a mean greater than 22 for thissample?• calculate the standard error of the mean (σ¯x=s√n)• calculate the z-score (z − s core =¯x−µσ¯x)• look up the z-score in a z table and determine the area under the curve that falls above thatpointThe answer ought to be less than 30. Why? Look up the answer in the table. Does everyoneunderstand how we can apply the same logic to the sample means as we do with the individualvalues?Recall that...• we always care about the population• we only have access to samples• we make inferences about populations from samples• we use probability to make educated guesses about samples given population informationHypothesis testing is merely the forma l means by which we make inferences about populationsfrom samples. We typically think about hypotheses in terms of population parameters. For exam-ple...• The mean hitting performance of Maj or League baseball players is .250• The mean IQ for the general population is 100• The mean body mass index for college students is 25kg/m2• The mean daily commute to work in the US was 24.3 minutes (2003)We use these as hypotheses - educated guesses - as to the values we are likely to observe in oursamples. Unfortunately, these hypotheses do not provide us with much to work with in terms ofwhat we typically do in science. We manipulate things (people, rodents, etc) to see whether thatmanipulation produces the change we expect. Consider the fo llowing manipulations...• weight-loss programs• medication to treat problem X• exercise programs• social welfare programs• vaccines• natural disasters32.5 State the hypotheses and define the critical regionWe study these types of manipulations to better understand the world and how humans exist withinthe world. Our typical use of the hypothesis in the f orm of the null hypothesis or H0. The nullhypothesis would be stated in the following way for the manipulations above:• a weight-loss program would not be different from no program• a medication to treat problem X would be no different t han no medication• an exercise progra m would produce no different outcomes from a non-exercise program• social welfare programs …