Sampling Distributions and Hypothesis Testing

2 Major Points An example An example Sampling distribution Sampling distribution Hypothesis testing Hypothesis testing  The null hypothesis  Test statistics and their distributions  The normal distribution and testing Important concepts Important concepts

3 The chance that something can happen may depend on… Fate Fate Probability Probability Gambling Gambling Error Error Likelihood Likelihood Expected variability in a sample Expected variability in a sample

4 Media Violence Does violent content in a video affect later behavior? Does violent content in a video affect later behavior?  Bushman (1998) Two groups of 100 subjects saw a video Two groups of 100 subjects saw a video  Violent video versus nonviolent video Then free associated to 26 homonyms with aggressive & nonaggressive forms. Then free associated to 26 homonyms with aggressive & nonaggressive forms.  e.g. cuff, mug, plaster, pound, sock Cont.

5 Media Violence Results Results  Number of aggressive free associates to the homonym as a function of video:  saw violent videomean = 7.10  saw 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?

6 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 study Convert to one-group study  People normally give 5.65 aggressive associates to homonyms. (a pop. parameter)  A group who watched violent videos give 7.10 aggressive associates. (a sample statistic)  Is this sufficiently more than expected to conclude that violent video has effect?

7 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?  Would we expect a similar kind of difference with a repeat of this experiment? Or...  Is the difference due to “sampling error?”

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

9 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.”

10 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.  See next slide.  Shows the kinds of means we expect to find when people don’t view a violent video.

11 In this case the sd is the standard error, not sample sd

12 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.  Our subjects’ responses were not like normal. Conclude that the violent video increased aggressive associations. Conclude that the violent video increased aggressive associations.

13 Hypothesis Testing A formal way of doing what we just did, compare sample statistic to population parameters A formal way of doing what we just did, compare sample statistic to population parameters  State research question  Assumptions  Null and Alternative Hypotheses  Sampling Distribution of Test Statistic  Decision Rule  Sample and Test Statistic  Decision Start with hypothesis that subjects are normal. Start with hypothesis that subjects are normal.  The null hypothesis: The is no difference…. The is no difference…. There is no association…. There is no association…. Find what normal subjects do. Find what normal subjects do. Compare our subjects to that standard. Compare our subjects to that standard.

14 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.

15 Chapter 8 Sampling Distributions Steps in Hypothesis Testing condensed: p. 151 Define the null hypothesis. (because we can rarely test the research H) Define the null hypothesis. (because we can rarely test the research H) Sample and decide what you would expect to find if the null hypothesis were true (given random sampling, what is the sampling distribution). Sample and decide what you would expect to find if the null hypothesis were true (given random sampling, what is the sampling distribution). Look at what you actually found and compare to the sampling distribution. What’s the probability of getting that statistic by chance? Look at what you actually found and compare to the sampling distribution. What’s the probability of getting that statistic by chance? Reject the null if what you found is not what you expected given the sampling distribution. Reject the null if what you found is not what you expected given the sampling distribution.

16 Important Concepts Concepts critical to hypothesis testing Concepts critical to hypothesis testing  Decision  Type I error  Type II error  Critical values  One- and two-tailed tests

17Decisions 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.  Type I error Reject the null hypothesis when it is actually correct. Reject the null hypothesis when it is actually correct.  Type II error Retain the null hypothesis when it is actually false. Retain the null hypothesis when it is actually false.

18 Type I Errors Assume violent videos really have no effect on associations Assume 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 error This is a Type I error  Probability set at alpha (  )  usually at.05  usually at.05  Therefore, probability of Type I error =.05

19 Type II Errors Assume violent videos make a difference Assume violent videos make a difference Assume that we conclude they don’t Assume that we conclude they don’t This is also an error This is also an error  Probability 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.

20 REJECT THE NULL HYP. FAIL TO REJECT THE NULL HYP. NULL IS FALSE 1. Correct decision 4. TYPE II ERROR NULL IS TRUE 3. TYPE I ERROR 2. Correct decision Four possible outcomes when testing null hypotheses:

21 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.  Our test statistic has some value with p =.05  We reject when we exceed that value.  That value is the critical value.

22 One- and Two-Tailed Tests Two-tailed test rejects null when obtained value too extreme in either direction Two-tailed test rejects null when obtained value too extreme in either direction  Decide 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)  We only set aside one direction for rejection. Cont.

23 One- & Two-Tailed Example One-tailed test One-tailed test  Reject 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 test Two-tailed test  Reject null if violent video group had an extreme number of aggressive associates; either too many or too few.

24 Review Questions Define a sampling distribution. Define a sampling distribution. How would you create sampling distribution of mean number of aggressive associates if the null were true? How would you create 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.

25 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.