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T-scores

T-scores. The next adventure. 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

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T-scores

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  1. T-scores The next adventure... Unit 2: z, t, hyp, 2t

  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 Unit 2: z, t, hyp, 2t

  3. Review: Different Measures of Stand. Dev. * Calculate differently based on available information Unit 2: z, t, hyp, 2t

  4. If σx is known, do z-test Use σx to get measure of sampling error in distribution. If σx is notknown, do t-test Use ŝx to get measure of sampling error in distribution. Different Measures of Sampling Error Unit 2: z, t, hyp, 2t

  5. t-distributions vs. z-distributions Unit 2: z, t, hyp, 2t

  6. Comparing Frequency & Sampling Distributions (T1) Unit 2: z, t, hyp, 2t

  7. Practice Problem: Calculating t-test • Do college students sleep 8 hours per night? • Follow hypothesis testing steps: • State type of comparison • State null (H0) and alternative (HA) • Set standards: • State type of test (& critical values if doing by hand ) • E.g., t-critical (get from table in back of book) • Significance level you require (eg. α = .05) • 1 vs. 2 tailed test (we’ll always do 2-tail tests- more conservative) • Calculate statistic (e.g. get t-obtained) • State decision and explain in English. Unit 2: z, t, hyp, 2t

  8. Finding t-critical Unit 2: z, t, hyp, 2t

  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 Unit 2: z, t, hyp, 2t

  10. HW: Standard Deviation Calculation Unit 2: z, t, hyp, 2t

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

  12. HW: Hypothesis testing steps • Compare xbar and μ • Ho: μ = 45 Ha: μ  45 • α = .05, df = n-1 = 8, two-tailed test. tcritical = 2.306 • tobt = .471 • 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. Unit 2: z, t, hyp, 2t

  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? Unit 2: z, t, hyp, 2t

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

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

  16. Output part #2 tobtained By hand, it’s • pobt: 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. Note: There’s no t-critical when done with SPSS Unit 2: z, t, hyp, 2t

  17. Compare xbar and μ Ho: μ = 55 Ha: μ  55 α = .05, df = n-1 = 8, two-tailed test. tobt = 1.492, pobt = .174 Retain Ho. Average car speed (M=58.89) does not differ significantly from 55 mph speed limit, t(8) = 1.492, n.s. Hypothesis Testing Steps Unit 2: z, t, hyp, 2t

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

  19. Learning Check • As tobt increases, we become more likely to ___ Ho. • If the sample size increases tobt will _____ and tcritical will ______ • If the difference between xbar and μ increases • sampling error will ______ • tcritical will _______ • tobtained will _______ • ŝxbar will _______ • you become _____ likely to reject the Ho Unit 2: z, t, hyp, 2t

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

  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. Unit 2: z, t, hyp, 2t

  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… • decide there’s a burglar when there is not. Type I Error • decide there’s no burglar when there is. Type II Error Unit 2: z, t, hyp, 2t

  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…. • Conclude participants don’t differ from average, but in reality the training DOES improve performance. Type II error • Conclude participants do better than average, but in reality the training does NOT improve performance. Type I error Unit 2: z, t, hyp, 2t

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

  25. You guess this… Ho: μ=55 Ha: μ>55 …but this is actually true. tcrit β 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. Unit 2: z, t, hyp, 2t

  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 H0 • 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 Unit 2: z, t, hyp, 2t

  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: x 2 3 4 3 2 2 2 1 • μ = • σ = • ŝx = • Ŝxbar= • M = • Mean Difference = • tcrit = • tobt = • pobt = • d = Unit 2: z, t, hyp, 2t

  28. Unit 2: z, t, hyp, 2t

  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: x 2 3 4 3 2 2 2 1 • μ = 3 • σ =??? • ŝx = .916 • Ŝxbar= .324 • M = 2.38 • Mean Diff. = -.625 • tcrit = ± 2.365 • tobt = -1.930 • pobt = .095 • d = inappropriate Unit 2: z, t, hyp, 2t

  30. #2: Hypothesis Testing Steps Unit 2: z, t, hyp, 2t

  31. #2: Hypothesis Testing Steps • Cf. M and μ. • Ho: μ = 3 Ha: μ≠ 3 • 2-tailed, α = .05, df=7 • tobt = -1.930, pobt = .095 • 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. Unit 2: z, t, hyp, 2t

  32. #3 Sketch the distribution, including regions of rejection, tcritical and tobtained. • #4 What type of decision error is possible here? • #5 Pretend you had a significant result – calculate d. Unit 2: z, t, hyp, 2t

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