Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell Chapter 12 Hypothesis Tests: One Sample Mean © 2003 Brooks/Cole Publishing Company/ITP

2Chapter 12 Hypothesis Tests: One Sample Mean Major Points An exampleAn example Sampling distribution of the meanSampling distribution of the mean Testing hypotheses: sigma knownTesting hypotheses: sigma known XAn example Testing hypotheses: sigma unknownTesting hypotheses: sigma unknown XAn example Cont.

3Chapter 12 Hypothesis Tests: One Sample Mean Major Points--cont. Factors affecting the testFactors affecting the test Measuring the size of the effectMeasuring the size of the effect Confidence limitsConfidence limits

4Chapter 12 Hypothesis Tests: One Sample Mean An Example Returning to the effect of violent videos on subsequent behaviorReturning to the effect of violent videos on subsequent behavior The first of several modifications of Bushman study from Chapter 8The first of several modifications of Bushman study from Chapter 8

5Chapter 12 Hypothesis Tests: One Sample Mean Media Violence Does violent content in a video affect subsequent responding?Does violent content in a video affect subsequent responding? 100 subjects saw a video containing considerable violence.100 subjects saw a video containing considerable violence. Then free associated to 26 homonyms that had an aggressive & nonaggressive form.Then free associated to 26 homonyms that had an aggressive & nonaggressive form. Xe.g. cuff, mug, plaster, pound, sock Cont.

6Chapter 12 Hypothesis Tests: One Sample Mean Media Violence--cont. ResultsResults XMean number of aggressive free associates = 7.10 Assume we know that without aggressive video the mean would be 5.65, and the standard deviation = 4.5Assume we know that without aggressive video the mean would be 5.65, and the standard deviation = 4.5  These are parameters (  and  Is 7.10 enough larger than 5.65 to conclude that video affected results?Is 7.10 enough larger than 5.65 to conclude that video affected results?

7Chapter 12 Hypothesis Tests: One Sample Mean Sampling Distribution of the Mean Formal solution to example given in Chapter 8.Formal solution to example given in Chapter 8. We need to know what kinds of sample means to expect if video has no effect.We need to know what kinds of sample means to expect if video has no effect.  i. e. What kinds of means if  = 5.65 and  = 4.5? XThis is the sampling distribution of the mean. Cont.

8Chapter 12 Hypothesis Tests: One Sample Mean Sampling Distribution of the Mean--cont. In Chapter 8 we saw exactly what this distribution would look like.In Chapter 8 we saw exactly what this distribution would look like. It is called Sampling Distribution of the Mean.It is called Sampling Distribution of the Mean. XWhy? XSee next slide. Cont.

9Chapter 12 Hypothesis Tests: One Sample Mean Cont.

10Chapter 12 Hypothesis Tests: One Sample Mean Sampling Distribution of the Mean--cont. The sampling distribution of the mean depends onThe sampling distribution of the mean depends on XMean of sampled population Why?Why? XSt. dev. of sampled population Why?Why? XSize of sample Why?Why? Cont.

11Chapter 12 Hypothesis Tests: One Sample Mean Sampling Distribution of the mean--cont. Shape of the sampling distributionShape of the sampling distribution XApproaches normal Why?Why? XRate of approach depends on sample size Why?Why? Basic theoremBasic theorem XCentral limit theorem

12Chapter 12 Hypothesis Tests: One Sample Mean Central Limit Theorem Given a population with mean =  and standard deviation = , the sampling distribution of the mean (the distribution of sample means) has a mean = , and a standard deviation =  /  n. The distribution approaches normal as n, the sample size, increases.Given a population with mean =  and standard deviation = , the sampling distribution of the mean (the distribution of sample means) has a mean = , and a standard deviation =  /  n. The distribution approaches normal as n, the sample size, increases.

13Chapter 12 Hypothesis Tests: One Sample MeanDemonstration Let population be very skewedLet population be very skewed Draw samples of 3 and calculate meansDraw samples of 3 and calculate means Draw samples of 10 and calculate meansDraw samples of 10 and calculate means Plot meansPlot means Note changes in means, standard deviations, and shapesNote changes in means, standard deviations, and shapes Cont.

14Chapter 12 Hypothesis Tests: One Sample Mean Parent Population Cont.

15Chapter 12 Hypothesis Tests: One Sample Mean Sampling Distribution n = 3 Cont.

16Chapter 12 Hypothesis Tests: One Sample Mean Sampling Distribution n = 10 Cont.

17Chapter 12 Hypothesis Tests: One Sample MeanDemonstration--cont. Means have stayed at 3.00 throughout-- except for minor sampling errorMeans have stayed at 3.00 throughout-- except for minor sampling error Standard deviations have decreased appropriatelyStandard deviations have decreased appropriately Shapes have become more normal--see superimposed normal distribution for referenceShapes have become more normal--see superimposed normal distribution for reference

18Chapter 12 Hypothesis Tests: One Sample Mean Testing Hypotheses:  known H 0 :  = 5.65H 0 :  = 5.65 H 1 :  5.65  (Two-tailed)H 1 :  5.65  (Two-tailed) Calculate p (sample mean) = 7.10 if  = 5.65Calculate p (sample mean) = 7.10 if  = 5.65 Use z from normal distributionUse z from normal distribution Sampling distribution would be normalSampling distribution would be normal

19Chapter 12 Hypothesis Tests: One Sample Mean Using z To Test H 0 Calculate zCalculate z If z > , reject H 0If z > , reject H > > 1.96 XThe difference is significant. Cont.

20Chapter 12 Hypothesis Tests: One Sample Meanz--cont. Compare computed z to histogram of sampling distributionCompare computed z to histogram of sampling distribution The results should look consistent.The results should look consistent. Logic of testLogic of test XCalculate probability of getting this mean if null true. XReject if that probability is too small.

21Chapter 12 Hypothesis Tests: One Sample Mean Testing When  Not Known Assume same example, but  not knownAssume same example, but  not known Can’t substitute s for  because s more likely to be too smallCan’t substitute s for  because s more likely to be too small XSee next slide. Do it anyway, but call answer tDo it anyway, but call answer t Compare t to tabled values of t.Compare t to tabled values of t.

22Chapter 12 Hypothesis Tests: One Sample Mean Sampling Distribution of the Variance Population variance = n = 5 10,000 samples 58.94% <

23Chapter 12 Hypothesis Tests: One Sample Mean t Test for One Mean Same as z except for s in place of .Same as z except for s in place of . For Bushman, s = 4.40For Bushman, s = 4.40

24Chapter 12 Hypothesis Tests: One Sample Mean Degrees of Freedom Skewness of sampling distribution of variance decreases as n increasesSkewness of sampling distribution of variance decreases as n increases t will differ from z less as sample size increasest will differ from z less as sample size increases Therefore need to adjust t accordinglyTherefore need to adjust t accordingly df = n - 1df = n - 1 t based on dft based on df

25Chapter 12 Hypothesis Tests: One Sample Mean t Distribution

26Chapter 12 Hypothesis Tests: One Sample MeanConclusions With n = 100, t = 1.98With n = 100, t = 1.98 Because t = 3.30 > 1.98, reject H 0Because t = 3.30 > 1.98, reject H 0 Conclude that viewing violent video leads to more aggressive free associates than normal.Conclude that viewing violent video leads to more aggressive free associates than normal.

27Chapter 12 Hypothesis Tests: One Sample Mean Factors Affecting t Difference between sample and population meansDifference between sample and population means Magnitude of sample varianceMagnitude of sample variance Sample sizeSample size

28Chapter 12 Hypothesis Tests: One Sample Mean Factors Affecting Decision Significance level Significance level  One-tailed versus two-tailed testOne-tailed versus two-tailed test

29Chapter 12 Hypothesis Tests: One Sample Mean Size of the Effect We know that the difference is significant.We know that the difference is significant. XThat doesn’t mean that it is important. Population mean = 5.65, Sample mean = 7.10Population mean = 5.65, Sample mean = 7.10 Difference is nearly 1.5 words, or 25% more violent words than normal.Difference is nearly 1.5 words, or 25% more violent words than normal. Cont.

30Chapter 12 Hypothesis Tests: One Sample Mean Effect Size (cont.) Later we will express this in terms of standard deviations.Later we will express this in terms of standard deviations. X1.45 units is 1.45/4.40 = 1/3 of a standard deviation.

31Chapter 12 Hypothesis Tests: One Sample Mean Confidence Limits on Mean Sample mean is a point estimateSample mean is a point estimate We want interval estimateWe want interval estimate  Probability that interval computed this way includes  = 0.95

32Chapter 12 Hypothesis Tests: One Sample Mean For Our Data

33Chapter 12 Hypothesis Tests: One Sample Mean Confidence Interval The interval does not include the population mean without a violent videoThe interval does not include the population mean without a violent video Consistent with result of t test.Consistent with result of t test. Confidence interval and effect size tell us about the magnitude of the effect.Confidence interval and effect size tell us about the magnitude of the effect. What can we conclude from confidence interval?What can we conclude from confidence interval?

34Chapter 12 Hypothesis Tests: One Sample Mean Review Questions What is the sampling distribution of the mean?What is the sampling distribution of the mean? Why do we need it?Why do we need it? Why did I assume that the population mean was 5.65 and its st. dev. was 0.45?Why did I assume that the population mean was 5.65 and its st. dev. was 0.45? What is the problem when the population variance (or st. dev.) is not known?What is the problem when the population variance (or st. dev.) is not known? Cont.

35Chapter 12 Hypothesis Tests: One Sample Mean Review Questions--cont. What does the shape of the sampling distribution of the variance have to do with anything?What does the shape of the sampling distribution of the variance have to do with anything? How does t differ from zHow does t differ from z What are degrees of freedom (df )?What are degrees of freedom (df )? What factors affect the value of t ?What factors affect the value of t ? Cont.

36Chapter 12 Hypothesis Tests: One Sample Mean Review Questions--cont. What factors affect our conclusions?What factors affect our conclusions? What do effect sizes tell us?What do effect sizes tell us? What is the difference between confidence limits and a confidence interval?What is the difference between confidence limits and a confidence interval? How do we interpret a computed confidence interval?How do we interpret a computed confidence interval?