1. Unit 4: Statistical Significance Lesson 2: Mean Differences - The Single Sample t-test EDER 6010: Statistics for Educational Research Dr. J. Kyle Roberts University of North Texas Next Slide

2. Mean Differences IQ Scores Next Slide 100 115 85 μ IQ = 100 σ IQ = 15

3. The Single Sample t-test “Does my sample mean differ statistically from the population mean?” “Are the sample and population means statistically significantly different from each other?” CI 95% = 105 ± 6.57 CI 95% = 105 ± 1.31 Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 20 SD = 3 Mean = 100 Mean = 105 Population Sample Study 2

4. Computing the ‘t’ statistic Range from - ∞ to +∞ What does a t = 0 mean? Next Slide

5. Alpha and the Null Hypothesis “ α = the point at which we reject our null hypothesis” 1 – α = Confidence Interval The probability of rejecting a Null Hypothesis when it is true CI 95% : α = .05 Next Slide

6. Critical Values for the t Distribution 95% Fail to Reject Area Next Slide -1.96 SD +1.96 SD 5% Reject Area 5% Reject Area

7. Critical Values for the t Distribution μ IQ = 100 σ IQ = 15 Next Slide

8. t Distribution Chart df = n-1 Level of significance = α t-critical for our study = 2.110 Page 637 in Hinkle et al. 4 th ed Page 638 in 5 th ed Next Slide

9. Determining Statistical Significance -2.110 t-critical 2.110 t = 4.72 We reject the hypothesis that our sample mean equals the population mean Next Slide -1.96 SD +1.96 SD

10. Comparing Studies t-crit = 2.093 Fail to reject the Null Hypothesis Reject the Null Hypothesis Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 20 SD = 3 Mean = 100 Mean = 105 Population Sample Study 2 CI 95% = 105 ± 6.57 SEM = 3.35 t-calc = 1.49 CI 95% = 105 ± 1.31 SEM = 0.67 t-calc = 7.46

11. Comparing Studies (again) CI 95% = 105 ± 6.57 CI 95% = 105 ± 0.89 SEM = 3.35 t-calc = 1.49 SEM = 1.36 t-calc = 3.68 t-crit = 2.093 Fail to reject the Null Hypothesis Reject the Null Hypothesis t-crit = 1.980 Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 121 SD = 15 Mean = 100 Mean = 105 Population Sample Study 2

12. Comparing Studies (one last time) CI 95% = 105 ± 6.57 CI 95% = 115 ± 6.57 SEM = 3.35 t-calc = 1.49 SEM = 3.35 t-calc = 4.48 Fail to reject the Null Hypothesis Reject the Null Hypothesis t-crit = 2.093 Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 20 SD = 15 Mean = 100 Mean = 115 Population Sample Study 2

13. Factors Affecting t-calc 1. The magnitude of the difference between the two means 2. The “spreadoutness” of the scores 3. The number of people in the study 4. The level at which α is set Next Slide

14. Running a Single Sample t-test in SPSS Analyze  Compare Means  One-Sample T Test Next Slide

15. Running a Single Sample t-test in SPSS (cont.) t-crit=2.365 Next Slide

16. Determining Statistical Significance by Alpha α is also: p-critical If: p-calc < p-crit, Reject the H 0 Next Slide α = .05 -2.365 t-critical 2.365 α = .05 p-calc

17. Determining Statistical Significance Means are not different statistically Fail to Reject the Null Hypothesis t-calc ≤ t-crit p-calc ≥ p-crit Means are different statistically Reject the Null Hypothesis t-calc > t-crit p-calc < p-crit Next Slide Statistically Significant Not Statistically Significant

18. Unit 4: Statistical Significance Lesson 2: Mean Differences - The Single Sample t-test EDER 6010: Statistics for Educational Research Dr. J. Kyle Roberts University of North Texas