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Sampling Distributions, Hypothesis Testing and One-sample Tests

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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.

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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?

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Sampling Distribution of the Mean 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.

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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.

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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

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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.

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Demonstration 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.

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Parent Population Cont.

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Sampling Distribution n = 3 Cont.

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Sampling Distribution n = 10 Cont.

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Demonstration--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

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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.

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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.

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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

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Decisions 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.

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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

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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.

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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.

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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.

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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.

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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

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Using z To Test H 0 Calculate zCalculate z If z > + 1.96, reject H 0If z > + 1.96, reject H 0 3.22 > 1.963.22 > 1.96 XThe difference is significant. Cont.

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z--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.

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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.

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Sampling Distribution of the Variance 138.89 Population variance = 138.89 n = 5 10,000 samples 58.94% < 138.89

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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

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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

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t Distribution

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Conclusions With n = 100, t.05 99 = 1.98With n = 100, t.05 99 = 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.

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Factors Affecting t Difference between sample and population meansDifference between sample and population means Magnitude of sample varianceMagnitude of sample variance Sample sizeSample size

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Factors Affecting Decision Significance level Significance level One-tailed versus two-tailed testOne-tailed versus two-tailed test

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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.

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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.

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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

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For Our Data

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Confidence Interval The interval does not include 5.65--the population mean without a violent videoThe interval does not include 5.65--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?

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