Univariate Statistics PSYC*6060

1. Univariate Statistics PSYC*6060 Class 2 Peter Hausdorf University of Guelph

2. Agenda • Review of first class • Howell Chapter 3 • Standard distributions exercise • Howell Chapter 4 • Block exercise • Hypothesis testing group work

3. Howell - Chapter 3 • Probabilities • Standard normal distributions • Standard scores

4. Probabilities - Education in Guelph

5. Another Example - Diffusion of Innovation % of consumers in each group adopting the product 13.5% 34% 34% 16% 2.5% Early Adopters Early Majority Late Majority Laggards Innovators Time 1997 1999 2001

6. Why are distributions useful? • Understanding the distribution allows us to interpret results/scores better • The distribution can help us to predict outcomes • Allows us to compare scores on instruments with different metrics • Used as a basis for most statistical tests

7. Standard Normal Distribution f(X) .40 0.3413 0.3413 0.1359 0.1359 0.0228 0.0228 0 -2 -1 0 1 2

8. Z = X - X SD Standard Scores P = nL x 100 N • Percentiles • z scores • T scores • CEEB scores T = (Z x 10)+50 A = (Z x 100)+500

9. Howell - Chapter 4 • Sampling distribution of the mean • Hypothesis testing • The Null hypothesis • Testing hypotheses with the normal distribution • Type I and II errors

10. Sampling distribution of the mean • Standard deviation of distribution reflects variability in sample statistic over repeated trials • Distribution of means of an infinite number of random samples drawn under certain specified conditions

11. Hypothesis testing • Establish research hypothesis • Obtain random sample • Establish null hypothesis • Obtain sampling distribution • Calculate probability of mean at least as large as sample mean • Make a decision to accept or reject null

12. The Null Hypothesis • We can never prove something to be true but we can prove something to be false • Provides a good starting point for any statistical test • If results don’t allow us to reject the null hypothesis then we have an inconclusive result

13. f(X) .40 0.3413 0.3413 0.1359 0.1359 0.0228 0.0228 0 -2 -1 0 1 2 Testing hypotheses using the normal distribution = 25 : = 5 F X = 32 N = 100 X - : Z = F N 32 - 25 Z = .5 Z = 14, p<.0001, Sig.

14. Type I error (alpha) • Is the probability of rejecting the null hypothesis when it is true • Border Collies - concluding that they are smarter than other dogs based on our study when in reality they are not • Relates to the rejection region we set (e.g. 5%, 1%)

15. Type II error • Is the probability of failing to reject the null hypothesis when it is false • Border Collies - concluding that they are not smarter than other dogs based on our study when in reality they are • Difficult to estimate given that we don’t know the distribution of data for our research hypothesis

16. Relationship between Type I and Type II Errors • The relationship is dynamic • The more stringent our rejection region the more we minimize Type I errors but the more we open ourselves up to Type II errors • Which error you want to minimize depends on the situation

17. f(X) .40 0 -2 -1 0 1 2 Relationship between Type I and Type II Errors 5% = 1.64 All Dogs 1% = 1.96 Type I Error Border Collies Type II Error -2 -1 1 0 2

18. Decision Making True State of the World Decision Reject H Fail to reject H H True Type I error p = alpha Correct decision p = 1 - alpha H False Correct decision p = (1 - beta) = power Type II error p = beta 0 0 0 0

19. One Versus Two Tailed Depends on your hypothesis going in. If you have a direction then can go with one tailed but if not then go with two tailed. Either way you have to respect the alpha level you have set for yourself.