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 Innovators Early Adopters Early Majority Late Majority Laggards Time 199719992001 2.5% 13.5%34% 16%

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 0.0228 0.1359 0.3413 0.1359 0.0228 0-221 f(X) 0.40

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

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. Testing hypotheses using the normal distribution 0.0228 0.1359 0.3413 0.1359 0.0228 0-221 f(X) 0.40 : = 25 = 5 X = 32 F Z = X - : F Z = 32 - 25.5 Z = 14, p<.0001, Sig. N = 100 N

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. 0-2 2 1 f(X) 0.40 0 -2 2 1 5% = 1.64 1% = 1.96 Type I Error Type II Error All Dogs Border Collies Relationship between Type I and Type II Errors

18. Decision Making 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 Decision Reject H Fail to reject H 0 0 True State of the World

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.