Sampling Distributions, Hypothesis Testing and One-sample Tests

1. Sampling Distributions, Hypothesis Testing and One-sample Tests

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

3. Media Violence--cont. • Results • Mean number of aggressive free associates = 7.10 • Assume we know that without aggressive video the mean would be 5.65, and the standard deviation = 4.5 • These are parameters (m and s) • Is 7.10 enough larger than 5.65 to conclude that video affected results?

4. Sampling Distribution of the Mean • We need to know what kinds of sample means to expect if video has no effect. • i. e. What kinds of means if m = 5.65 and s = 4.5? • This is the sampling distribution of the mean. Cont.

5. Cont.

6. Sampling Distribution of the Mean--cont. • The sampling distribution of the mean depends on • Mean of sampled population • Why? • St. dev. of sampled population • Why? • Size of sample • Why? Cont.

7. Sampling Distribution of the mean--cont. • Shape of the sampling distribution • Approaches normal • Why? • Rate of approach depends on sample size • Why? • Basic theorem • Central limit theorem

8. Central Limit Theorem • Given a population with mean = m and standard deviation = s, the sampling distribution of the mean (the distribution of sample means) has a mean = m, and a standard deviation = s /n. The distribution approaches normal as n, the sample size, increases.

9. Demonstration • Let population be very skewed • Draw samples of 3 and calculate means • Draw samples of 10 and calculate means • Plot means • Note changes in means, standard deviations, and shapes Cont.

10. Parent Population Cont.

11. Sampling Distribution n = 3 Cont.

12. Sampling Distribution n = 10 Cont.

13. Demonstration--cont. • Means have stayed at 3.00 throughout--except for minor sampling error • Standard deviations have decreased appropriately • Shapes have become more normal--see superimposed normal distribution for reference

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

15. The Null Hypothesis • 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 we usually want to reject.

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

17. Decisions • 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. • Type II error • Retain the null hypothesis when it is actually false.

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

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

20. Critical Values • These represent the point at which we decide to reject null hypothesis. • 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.

21. One- and Two-Tailed Tests • 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) • We only set aside one direction for rejection. Cont.

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

23. Testing Hypotheses: s known • H0: m = 5.65 • H1: m  5.65(Two-tailed) • Calculate p (sample mean) = 7.10 if m = 5.65 • Use z from normal distribution • Sampling distribution would be normal

24. Using z To Test H0 • Calculate z • If z> + 1.96, reject H0 • 3.22 > 1.96 • The difference is significant. Cont.

25. z--cont. • Compare computed z to histogram of sampling distribution • The results should look consistent. • Logic of test • Calculate probability of getting this mean if null true. • Reject if that probability is too small.

26. Testing When s Not Known • Assume same example, but s not known • Can’t substitute s for s because s more likely to be too small • See next slide. • Do it anyway, but call answer t • Compare t to tabled values of t.

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

28. t Test for One Mean • Same as z except for s in place of s. • For Bushman, s = 4.40

29. Degrees of Freedom • Skewness of sampling distribution of variance decreases as n increases • t will differ from z less as sample size increases • Therefore need to adjust t accordingly • df = n - 1 • t based on df

30. t Distribution

31. Conclusions • With n = 100, t.0599 = 1.98 • Because t = 3.30 > 1.98, reject H0 • Conclude that viewing violent video leads to more aggressive free associates than normal.

32. Factors Affecting t • Difference between sample and population means • Magnitude of sample variance • Sample size

33. Factors Affecting Decision • Significance level a • One-tailed versus two-tailed test

34. Size of the Effect • We know that the difference is significant. • That doesn’t mean that it is important. • Population mean = 5.65, Sample mean = 7.10 • Difference is nearly 1.5 words, or 25% more violent words than normal. Cont.

35. Effect Size (cont.) • Later we will express this in terms of standard deviations. • 1.45 units is 1.45/4.40 = 1/3 of a standard deviation.

36. Confidence Limits on Mean • Sample mean is a point estimate • We want interval estimate • Probability that interval computed this way includes m = 0.95

37. For Our Data

38. Confidence Interval • The interval does not include 5.65--the population mean without a violent video • Consistent with result of t test. • Confidence interval and effect size tell us about the magnitude of the effect. • What can we conclude from confidence interval?