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Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 1 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 Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 2 Review: Different Measures of Stand. Dev. * Calculate differently based on available information Have all the scores in a population Have only scores in a sample, want to estimate variability in population Dr. Sinn, PSYC 301 x x n 2 E.g., SAT scores (ETS has every single score). 2 x E.g., hours of sleep students in this class slept last night (Need to adjust because you’ve only got sample data.) Unit 2: z, t, hyp, 2t sˆx n x 2 2 x n 1 n p. 3 Different Measures of Sampling Error • If σx is known, do z-test • If σx is not known, do t-test • Use σx to get measure of sampling error in distribution. • Use ŝx to get measure of sampling error in distribution. x x sˆx sˆx n so... x t sˆx n so... x z x Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 4 t-distributions vs. z-distributions Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 5 Comparing Frequency & Sampling Distributions (T1) Frequency D-z Sampling D – z Sampling D - t x ’s xbars xbars x x x Amt. of Variab. + Meas. of Variab. x x sx Have Compare Formula Dr. Sinn, PSYC 301 z x z x x Unit 2: z, t, hyp, 2t x t sˆx p. 6 Practice Problem: Calculating t-test • Do college students sleep 8 hours per night? • Follow hypothesis testing steps: 1. State type of comparison 2. State null (H0) and alternative (HA) 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. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 7 Finding t-critical Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 8 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 Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 9 HW: Standard Deviation Calculation Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 10 HW: T-Calculation • SD = 10.6066 • SE = 3.536 • t = (46.67-45) / 3.536 = .4781 Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 11 HW: Hypothesis testing steps 1. Compare xbar and μ 2. Ho: μ = 45 Ha: μ 45 3. α = .05, df = n-1 = 8, two-tailed test. tcritical = 2.306 4. tobt = .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. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 12 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? Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 13 SPSS Steps Go to Compare Means Pick variable Enter the speeds of cars you clocked. Dr. Sinn, PSYC 301 Set to μ Unit 2: z, t, hyp, 2t p. 14 Output part #1 Number of cars you measured (sample size). Average speed of these cars (sample mean). One-Sample Statistics SPEED N 9 Mean 58.89 x Std. Deviation 7.817 Std. Error Mean 2.606 ŝ x ŝ x Standard error of the mean – the typical difference we’d expected sampling error to cause. Standard deviation of these speeds. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 15 Output part #2 One-Sample Test Test Value = 55 tobtained By hand, it’s x t sˆx SPEED t 1.492 Sig. (2-tailed) .174 df 8 x •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. Dr. Sinn, PSYC 301 Mean Difference 3.89 95% Confidenc e Int erval of t he Difference Lower Upper -2. 12 9.90 Unit 2: z, t, hyp, 2t Note: There’s no tcritical when done with SPSS p. 16 Hypothesis Testing Steps 1. Compare xbar and μ 2. Ho: μ = 55 Ha: μ 55 3. α = .05, df = n-1 = 8, two-tailed test. 4. tobt = 1.492, pobt = .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. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 17 Same test, different outcome • What if we had measured slightly different speeds? • 50,60,65,55,65,60,55,75,65 • In this case, we’d reject the Ho. • Speeds appear to exceed 55 mph, t(8) = 2.475, p.05 • What happens to μ? xbar? One-Sample Statistics N SPEED 9 Mean 61.11 Std. Deviation 7.407 Std. Error Mean 2.469 One-Sam ple Test Test Value = 55 SPEED Dr. Sinn, PSYC 301 t 2.475 df 8 Sig. (2-tailed) .038 Unit 2: z, t, hyp, 2t Mean Difference 6.11 95% Confidenc e Int erval of t he Difference Lower Upper .42 11.80 p. 18 Learning Check 1. As tobt increases, we become more likely to ___ Ho. 2. If the sample size increases tobt will _____ and tcritical will ______ 3. If the difference between xbar and μ increases a. b. c. d. e. sampling error will ______ tcritical will _______ tobtained will _______ ŝxbar will _______ you become _____ likely to reject the Ho Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 19 Learning Check 1. A researcher compares the number of workdays missed for employees who are depressed versus the companywide 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 ŝx of 1.5 or 3? Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 20 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. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 21 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 Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 22 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 Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 23 Graph of Type I Error – α When rejecting Ho, you may commit a Type I error. (Wrongly concluding cars DO NOT average 55 mph.) But this is actually true. Ho: μ=55 Ha: μ>55 α You guess this. α tcrit tcrit So α is the chance of making a Type I error. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 24 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.) You guess this… Ho: μ=55 Ha: μ>55 …but this is actually true. β tcrit So β is the chance of making a Type II error. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 25 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 This is x d = .5+ moderate effect size d standard d = .8+ large effect size sˆx deviation, not standard error Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 26 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: μ= Mean Difference = σ= tcrit = ŝx = tobt = Ŝxbar= pobt = M= Dr. Sinn, PSYC 301 x 2 3 4 3 2 2 2 1 d= Unit 2: z, t, hyp, 2t p. 27 One-Sample Statistics N anxiety 8 Std. Deviation .916 Mean 2.38 Std. Error Mean .324 One-Sample Test Test Value = 3 anxiety Dr. Sinn, PSYC 301 t -1.930 df 7 Unit 2: z, t, hyp, 2t Sig. (2-tailed) .095 Mean Difference -.625 p. 28 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: μ=3 Mean Diff. σ =??? tcrit = ± 2.365 ŝx = .916 tobt = -1.930 Ŝxbar= .324 d= M= Dr. Sinn, PSYC 301 2.38 = -.625 x 2 3 4 3 2 2 2 1 pobt = .095 inappropriate Unit 2: z, t, hyp, 2t p. 29 • #2: Hypothesis Testing Steps Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 30 #2: Hypothesis Testing Steps 1. Cf. M and μ. 2. Ho: μ = 3 Ha: μ ≠ 3 3. 2-tailed, α = .05, df=7 4. tobt = -1.930, pobt = .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. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 31 • #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. Dr. Sinn, PSYC 301 Unit 2: z, t, hyp, 2t p. 32