1. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 1

2. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 2 Overview of t-scores Very similar to z-scores –Provides way of judging how extreme a sample mean is –A bunch of t-scores form a t-distribution Done when σ is unknown Used for hypothesis testing: –Ex: You wonder if college students really get 8 hours of sleep Ho: μ = 8 (College students do get eight hours of sleep) Ha: μ  8 (College students don’t get eight hours of sleep) t-distribution provides foundation for t-test –can do by hand with table –can do on SPSS Key difference: t-test done when σ is unknown

3. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 3 Review: Different Measures of Stand. Dev. Have all the scores in a population E.g., SAT scores (ETS has every single score). Have only scores in a sample, want to estimate variability in population E.g., hours of sleep students in this class slept last night (Need to adjust because you’ve only got sample data.) * Calculate differently based on available information

4. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 4 Different Measures of Sampling Error If σ x is known, do z-test Use σ x to get measure of sampling error in distribution. If σ x is not known, do t-test Use ŝ x to get measure of sampling error in distribution.

5. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 5 t-distributions vs. z-distributions

6. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 6 Comparing Frequency & Sampling Distributions (T1) Frequency D-zSampling D – zSampling D - t Have x ’s x bars Compare Amt. of Variab. ++ Meas. of Variab. Formula

7. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 7 Practice Problem: Calculating t-test Do college students sleep 8 hours per night?Do college students sleep 8 hours per night? Follow hypothesis testing steps: 1.State type of comparison 2.State null (H 0 ) and alternative (H A ) 3.Set standards: a.State type of test (& critical values if doing by hand ) E.g., t-critical (get from table in back of book) b.Significance level you require (eg. α =.05) c.1 vs. 2 tailed test (we’ll always do 2-tail tests- more conservative) 4.Calculate statistic (e.g. get t-obtained) 5.State decision and explain in English.

8. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 8 Finding t-critical

9. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 9 Homework Problem College graduates score 35, 45, 30, 50, 60, 55, 60, 45, 40 on a critical thinking test. If normal people score 45 on the test, do college graduates score significantly better? Do hypothesis testing steps

10. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 10 HW: Standard Deviation Calculation

11. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 11 HW: T-Calculation SD = 10.6066 SE = 3.536 t = (46.67-45) / 3.536 =.4781

12. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 12 HW: Hypothesis testing steps 1.Compare x bar and μ 2.Ho: μ = 45 Ha: μ  45 3.α =.05, df = n-1 = 8, two-tailed test. t critical = 2.306 4.t obt =.471 5.Retain Ho. The hypothesis was not supported. College graduates did not score significantly better (M=46.67) on critical thinking (μ =45), t(8) =.471, n.s.

13. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 13 T-test Example: Speed The government claims cars traveling in front of your house average 55 mph. You think this is a load of…. That is, you think cars travel faster than this. You steal a police radar gun and clock nine cars, obtaining the following speeds: 45, 60, 65, 55, 65, 60, 50, 70, 60 What’s μ?

14. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 14 SPSS Steps Go to Compare Means Pick variable Set to μ Enter the speeds of cars you clocked.

15. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 15 Output part #1 Average speed of these cars (sample mean). Standard deviation of these speeds. Standard error of the mean – the typical difference we’d expected sampling error to cause. Number of cars you measured (sample size).

16. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 16 Output part #2 t obtained p obt : Proportion of time you’d see a difference of this size simply because of sampling error This value must fall below.05 to say we have a significant difference. By hand, it’s Note: There’s no t- critical when done with SPSS

17. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 17 Hypothesis Testing Steps 1.Compare x bar and μ 2.Ho: μ = 55 Ha: μ  55 3.α =.05, df = n-1 = 8, two-tailed test. 4.t obt = 1.492, p obt =.174 5.Retain Ho. Average car speed (M=58.89) does not differ significantly from 55 mph speed limit, t(8) = 1.492, n.s.

18. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 18 Same test, different outcome What if we had measured slightly different speeds? 50,60,65,55,65,60,55,75,65 What happens to μ? x bar ? In this case, we’d reject the Ho. Speeds appear to exceed 55 mph, t(8) = 2.475, p .05

19. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 19 Learning Check 1.As t obt increases, we become more likely to ___ Ho. 2.If the sample size increases t obt will _____ and t critical will ______ 3.If the difference between x bar and μ increases a.sampling error will ______ b.t critical will _______ c.t obtained will _______ d.ŝ xbar will _______ e.you become _____ likely to reject the Ho

20. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 20 Learning Check 1.A researcher compares the number of workdays missed for employees who are depressed versus the company- wide average of 6 days per year. a.Rejecting the Ho would mean what about depressed employees? b.Would you be more likely to reject Ho with a sample mean of 8 or 10? c.Would you be more likely to reject Ho with a ŝ x of 1.5 or 3?

21. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 21 Decision Errors Educated guesses can be wrong. Def: Drawing a false conclusion from an hypothesis test –Never know for sure if a difference is due just to sampling error or if it truly reflects a treatment effect. Two Types –Type I: Falsely rejecting null Seeing something that’s not there. False positive. –Type II: Falsely retaining null Missing something that is there. False negative.

22. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 22 Decision Errors – Example #1 “Is that a burglar or am I hearing things?” You hear a noise in your house and wonder if it means there’s a burglar in the house. The problem is that it could just be regular background noise (___________) or it really could mean something’s going on (____________). You’d make a mistake if you… a.decide there’s a burglar when there is not. Type I Error b.decide there’s no burglar when there is. Type II Error

23. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 23 Decision Errors – Example #2 “Did the training work or is this group of people just more talented than usual?” You implement a training program to improve job performance, and then compare the performance of trainees to average performance. You’d make a mistake if you…. a.Conclude participants don’t differ from average, but in reality the training DOES improve performance. Type II error b.Conclude participants do better than average, but in reality the training does NOT improve performance. Type I error

24. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 24 Graph of Type I Error – α When rejecting Ho, you may commit a Type I error. (Wrongly concluding cars DO NOT average 55 mph.) You guess this. Ha: μ>55 But this is actually true. Ho: μ=55 t crit αα So α is the chance of making a Type I error.

25. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 25 Graph of Type II Error – β When retaining Ho, you may commit a Type II error. (In this case, assuming cars DO average 55 mph.) So β is the chance of making a Type II error. Ho: μ=55 You guess this… t crit β Ha: μ>55 …but this is actually true.

26. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 26 Effect-size statistic: d Statistical vs. Practical Significance –Statistical Sig: Decides if difference is reliable (e.g., t-test) –Practical Sig: Decides if difference is big enough to be practically important –So, only do tests for practical significance if you get statistical significance first (i.e., if you reject the H 0 Effect size (d) –Def: Impact of IV on DV in terms of standard deviation units. –So, d=1 means the IV “raises” scores 1 full standard deviation. –d =.2+ small effect size –d =.5+ moderate effect size –d =.8+ large effect size This is standard deviation, not standard error

27. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 27 Practice: Meditation You suspect the anxiety level of people in your meditation class will differ from a score of 3 on a 1-5 anxiety self-assessment scale. #1: Do an SPSS analysis and then fill-in the following information: x23432221x23432221 μ = σ = ŝ x = Ŝ xbar = M = Mean Difference = t crit = t obt = p obt = d =

28. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 28

29. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 29 Practice: Meditation You suspect the anxiety level of people in your meditation class will differ from a score of 3 on a 1-5 anxiety self-assessment scale. #1: Do an SPSS analysis and then fill-in the following information: x23432221x23432221 μ = 3 σ =??? ŝ x =.916 Ŝ xbar =.324 M = 2.38 Mean Diff. = -.625 t crit = ± 2.365 t obt = -1.930 p obt =.095 d = inappropriate

30. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 30 #2: Hypothesis Testing Steps

31. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 31 #2: Hypothesis Testing Steps 1.Cf. M and μ. 2.Ho: μ = 3 Ha: μ ≠ 3 3.2-tailed, α =.05, df=7 4.t obt = -1.930, p obt =.095 5.Retain Ho. The hypothesis was not supported. The anxiety of those meditating (M=2.38) did not differ significantly from average anxiety (μ=3), t(7) = -1.930, n.s.

32. Dr. Sinn, PSYC 301Unit 2: z, t, hyp, 2t p. 32 #3 Sketch the distribution, including regions of rejection, t critical and t obtained. #4 What type of decision error is possible here? #5 Pretend you had a significant result – calculate d.