Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell Chapter 8 Sampling Distributions and Hypothesis Testing © 2003 Brooks/Cole Publishing Company/ITP

2Chapter 8 Sampling Distributions Major Points An exampleAn example Sampling distributionSampling distribution Hypothesis testingHypothesis testing XThe null hypothesis XTest statistics and their distributions XThe normal distribution and testing Important conceptsImportant concepts

3Chapter 8 Sampling Distributions Media Violence Does violent content in a video affect later behavior?Does violent content in a video affect later behavior? XBushman (1998) Two groups of 100 subjects saw a videoTwo groups of 100 subjects saw a video XViolent video versus nonviolent video Then free associated to 26 homonyms with aggressive & nonaggressive forms.Then free associated to 26 homonyms with aggressive & nonaggressive forms. Xe.g. cuff, mug, plaster, pound, sock Cont.

4Chapter 8 Sampling Distributions Media Violence ResultsResults XNumber of aggressive free associates to the homonym as a function of video: Xsaw violent videomean = 7.10 Xsaw nonviolent video mean = 5.65 Is this difference large enough to conclude that type of video affected results?Is this difference large enough to conclude that type of video affected results?

5Chapter 8 Sampling Distributions A Simplified Version of Study One-group study is easier to start with than two-group study.One-group study is easier to start with than two-group study. Convert to one-group studyConvert to one-group study XPeople normally give 5.65 aggressive associates to homonyms. (a pop. parameter) XA group who watched violent videos give 7.10 aggressive associates. (a sample statistic) XIs this sufficiently more than expected to conclude that violent video has effect?

6Chapter 8 Sampling Distributions What is the Question? Is the difference between 7.10 and 5.65 large enough to lead us to conclude that it is a real difference?Is the difference between 7.10 and 5.65 large enough to lead us to conclude that it is a real difference? XWould we expect a similar kind of difference with a repeat of this experiment? Or... XIs the difference due to “sampling error?”

7Chapter 8 Sampling Distributions Sampling Error The normal variability that we would expect to find from one sample to another, or one study to anotherThe normal variability that we would expect to find from one sample to another, or one study to another Random variability among observations or statistics that is just due to chanceRandom variability among observations or statistics that is just due to chance

8Chapter 8 Sampling Distributions How Could we Assess Sampling Error? Take many groups of 100 subjects who did not see a violent video.Take many groups of 100 subjects who did not see a violent video. Record the number of aggressive responses to 26 homonyms.Record the number of aggressive responses to 26 homonyms. Plot the distribution and record its mean and standard deviation.Plot the distribution and record its mean and standard deviation. This distribution is a “Sampling Distribution.”This distribution is a “Sampling Distribution.”

9Chapter 8 Sampling Distributions Sampling Distribution The distribution of a statistic over repeated sampling from a specified population.The distribution of a statistic over repeated sampling from a specified population. Possible result for this example.Possible result for this example. XSee next slide. XShows the kinds of means we expect to find when people don’t view a violent video.

10Chapter 8 Sampling Distributions

11Chapter 8 Sampling Distributions What Do We Conclude? When people don’t view violent video, they average between about 4.5 and 6.5 aggressive interpretations of homonyms.When people don’t view violent video, they average between about 4.5 and 6.5 aggressive interpretations of homonyms. Our violent video group averaged 7.10 aggressive interpretations.Our violent video group averaged 7.10 aggressive interpretations. XOur subjects’ responses were not like normal. Conclude that the violent video increased aggressive associations.Conclude that the violent video increased aggressive associations.

12Chapter 8 Sampling Distributions Hypothesis Testing A formal way of doing what we just didA formal way of doing what we just did Start with hypothesis that subjects are normal.Start with hypothesis that subjects are normal. XThe null hypothesis Find what normal subjects do.Find what normal subjects do. Compare our subjects to that standard.Compare our subjects to that standard.

13Chapter 8 Sampling Distributions The Null Hypothesis The hypothesis that our subjects came from a population of normal responders.The hypothesis that our subjects came from a population of normal responders. The hypothesis that watching a violent video does not change mean number of aggressive interpretations.The hypothesis that watching a violent video does not change mean number of aggressive interpretations. The hypothesis we usually want to reject.The hypothesis we usually want to reject.

14Chapter 8 Sampling Distributions Steps in Hypothesis Testing Define the null hypothesis.Define the null hypothesis. Decide what you would expect to find if the null hypothesis were true.Decide what you would expect to find if the null hypothesis were true. Look at what you actually found.Look at what you actually found. Reject the null if what you found is not what you expected.Reject the null if what you found is not what you expected.

15Chapter 8 Sampling Distributions Important Concepts Concepts critical to hypothesis testingConcepts critical to hypothesis testing XDecision XType I error XType II error XCritical values XOne- and two-tailed tests

16Chapter 8 Sampling DistributionsDecisions When we test a hypothesis we draw a conclusion; either correct or incorrect.When we test a hypothesis we draw a conclusion; either correct or incorrect. XType I error Reject the null hypothesis when it is actually correct.Reject the null hypothesis when it is actually correct. XType II error Retain the null hypothesis when it is actually false.Retain the null hypothesis when it is actually false.

17Chapter 8 Sampling Distributions Type I Errors Assume violent videos really have no effect on associationsAssume violent videos really have no effect on associations Assume we conclude that they do.Assume we conclude that they do. This is a Type I errorThis is a Type I error XProbability set at alpha (  )  usually at.05  usually at.05 XTherefore, probability of Type I error =.05

18Chapter 8 Sampling Distributions Type II Errors Assume violent videos make a differenceAssume violent videos make a difference Assume that we conclude they don’tAssume that we conclude they don’t This is also an error (Type II)This is also an error (Type II) XProbability denoted beta (  ) We can’t set beta easily.We can’t set beta easily. We’ll talk about this issue later.We’ll talk about this issue later. Power = (1 -  ) = probability of correctly rejecting false null hypothesis.Power = (1 -  ) = probability of correctly rejecting false null hypothesis.

19Chapter 8 Sampling Distributions Critical Values These represent the point at which we decide to reject null hypothesis.These represent the point at which we decide to reject null hypothesis. e.g. We might decide to reject null when (p|null) <.05.e.g. We might decide to reject null when (p|null) <.05. XOur test statistic has some value with p =.05 XWe reject when we exceed that value. XThat value is the critical value.

20Chapter 8 Sampling Distributions One- and Two-Tailed Tests Two-tailed test rejects null when obtained value too extreme in either directionTwo-tailed test rejects null when obtained value too extreme in either direction XDecide on this before collecting data. One-tailed test rejects null if obtained value is too low (or too high)One-tailed test rejects null if obtained value is too low (or too high) XWe only set aside one direction for rejection. Cont.

21Chapter 8 Sampling Distributions One- & Two-Tailed Example One-tailed testOne-tailed test XReject null if violent video group had too many aggressive associates Probably wouldn’t expect “too few,” and therefore no point guarding against it.Probably wouldn’t expect “too few,” and therefore no point guarding against it. Two-tailed testTwo-tailed test XReject null if violent video group had an extreme number of aggressive associates; either too many or too few.

22Chapter 8 Sampling Distributions Review Questions Define a sampling distribution.Define a sampling distribution. How would you create a sampling distribution of mean number of aggressive associates if the null were true?How would you create a sampling distribution of mean number of aggressive associates if the null were true? What is sampling error?What is sampling error? What does sampling error have to do with all of this?What does sampling error have to do with all of this? Cont.

23Chapter 8 Sampling Distributions Review Questions--cont. What are the steps in hypothesis testing?What are the steps in hypothesis testing? What is the probability we’d conclude violent videos cause aggression if they really don’t?What is the probability we’d conclude violent videos cause aggression if they really don’t? Distinguish between Type I and Type II errors.Distinguish between Type I and Type II errors. Distinguish between one-tailed and two- tailed tests.Distinguish between one-tailed and two- tailed tests.