1. HIM 3200 Chapters 8 & 9 Hypothesis Testing and T-Tests Dr. Burton

2. 6.5 120 0 BP Z 95 101 -1.65 -1.28 5% 40% 139 145 1.28 1.65

3. 6.12 500 0 xZxZ.05.45 665 1.65 x = 500 + (1.65)100 = 665 5% of 1,000,000 = 50,000

4. 6.12 500 0 xZxZ.10 628 1.28 x = 500 + (.25)100 = 525 x = 500 + (1.28)100 = 628 30% of 1,000,000 = 300,000.30 525.25

5. 6.12 500 0 xZxZ.05.45 665 1.65 x = 500 + (1.65)100 = 665 x = 500 + (-1.65)100 = 335 90% betweem 335 and 665.45 335 -1.65.05

6. 6.12 500 0 xZxZ.0668.50 628 1.28 Z = 350 – 500/100 = -1.50.0668 or about 7% would score less than 350.4332 350 -1.50

7. 6.19  = 200  = 25 < 165 = 165 -200/25 = -1.4 =.5 -.4192 =.0808 1 -.0808 =.9192

8. 7.2a 60 0 Height Z 5763 % t = x -  s / n 57 - 60 10 / 25 63 - 60 10 / 25 Z= -1.5 =.4332 Z= 1.5 =.4332.8664 = 86.8%

9. 7.2b 60 0 Height Z 58 % t = x -  s / n 58 - 60 10 / 25 Z = -1.0 =.5000 -.3413.1587 = 16%

10. 7.2c 60 0 0.5 Height Z 61 % t = x -  s / n 61 - 60 10 / 25 Z = 0.50 -.1915 =.3085 = 30.9%= 0.50

11. 7.16a 2400 0 2.0 Height Z 2500 % t = x -  s / n 2500 - 2400 400 / 64 Z = 2.0 -.4772 =.0228 = 2.3%= 0.50

12. 7.16b 2400 0 Height Z 23002500 % t = x -  s / n 2300 - 2400 400 / 64 2500 - 2400 400 / 64 Z= -2.0 =.4772 Z= 2.0 =.4772.9544 = 95.4%

13. 7.16c 2400 0 Height Z 2350 % t = x -  s / n 2350 - 2400 400 / 64 Z = -1.0 -.3413 =.1587 = 16%= 0.50

14. Hypothesis Testing Hypothesis: A statement of belief… Null Hypothesis, H 0 : …there is no difference between the population mean  and the hypothesized value  0. Alternative Hypothesis, H a : …reject the null hypothesis and accept that there is a difference between the population mean  and the hypothesized value  0.

15. Probabilities of Type I and Type II errors H 0 TrueH 0 False Accept H 0 Reject H 0 Type I Error Type II Error Correct results Correct results Truth Test result  1 -   1 -  H 0 True = statistically insignificant H 0 False = statistically significant Accept H 0 = statistically insignificant Reject H 0 = statistically significant Differences ab cd http://en.wikipedia.org/wiki/False_positive

16. -3 -2 0 123 SE Probability Distribution for a two-tailed test SE Magnitude of (X E – X C ) 1.96 SE X E < X C X E > X C  = 0.05 0.025

17. -3 -2 0 123 SE Probability Distribution for a one-tailed test SE Magnitude of (X E – X C ) 1.645 SE X E < X C X E > X C  = 0.05

18. Box 10 - 5 t = A Distance between the means Variation around the means

19. Box 10 - 5 t = A B Distance between the means Variation around the means

20. Box 10 - 5 t = A B C Distance between the means Variation around the means

21. t-Tests Students t-test is used if: –two samples come from two different groups. –e.g. A group of students and a group of professors Paired t-test is used if: –two samples from the sample group. –e.g. a pre and post test on the same group of subjects.

22. One-Tailed vs. Two Tailed Tests The Key Question: “Am I interested in the deviation from the mean of the sample from the mean of the population in one or both directions.” If you want to determine whether one mean is significantly from the other, perform a two-tailed test. If you want to determine whether one mean is significantly larger, or significantly smaller, perform a one-tailed test.

23. t-Test (Two Tailed) Independent Sample means x A - x B - 0 t = Sp [ ( 1/N A ) + ( 1/N B ) ] d f = N A + N B - 2

24. Independent Sample Means Sample A(A – Mean) 2 2634.34 2414.90 184.58 179.86 184.58 20.02 184.58 Mean = 20.14  A 2 = 2913 N = 7 (A – Mean) 2 = 72.86 Var = 12.14 s = 3.48 Sample B(B – Mean) 2 38113.85 261.77 2411.09 307.13 2228.41 Mean = 27.33  B 2 = 4656 N = 6 (B – Mean) 2 = 173.34 Var = 34.67 s = 5.89

25. Standard error of the difference between the means (SED) SED of  E -  C = s A 2 Estimate of the s B 2 N AN A N BN B + SED of x E - x C =  A 2  B 2 N AN A N BN B + Theoretical Population Sample

26. Pooled estimate of the SED (SEDp) 1 Estimate of the 1 N AN A N BN B + SEDp of x A - x B = Sp s 2 (n A -1) + s 2 (n B – 1) Sp= n A + n B - 2 12.14 ( 6 ) + 34.67 ( 5 ) Sp= 7 + 6 - 2 = 22.38 = 4.73

27. t-Test (Two Tailed) d f = N E + N C - 2 = 11 x A - x B - 0 t = Sp [ ( 1/N A ) + ( 1/N B ) ] 20.14 - 27.33 - 0 = 4.73 ( 1/7 ) + ( 1/6) = -2.73 Critical Value 95% = 2.201

28. One-tailed and two-tailed t-tests A two-tailed test is generally recommended because differences in either direction need to be known.

29. Paired t-test t paired = t p = d - 0 Standard error of d = ------------- d - 0 S d 2 N df = N - 1 d =  D/N  d 2 =  D 2 – (  D) 2 / N S d 2 =  d 2 / N - 1

30. Pre/post attitude assessment StudentBeforeAfterDifferenceD squared 1252839 22319-416 33034 416 4710 39 536 39 62226 416 71213 11 83047 17289 9516 11121 10149-525 Total171208  D = 37  D 2 = 511

31. Pre/post attitude assessment StudentBeforeAfterDifference D squared Total171208 37 511 t paired = t p = d - 0 Standard error of d = ------------- d - 0 S d 2 N d =  D/N N = 10  d 2 =  D 2 – (  D) 2 / N S d 2 =  d 2 / N - 1 = 37/10 = 3.7 = 511 - 1369/10 = 374.1 = 374.1 / 10 – 1 = 41.5667 = 3.7 / 2.0387 = 1.815 = 3.7 / 41.5667 / 10 = 3.7 / 4. 15667 df = N – 1 = 9 0.05 > 1.833

32. Probabilities of Type I and Type II errors H 0 TrueH 0 False Accept H 0 Reject H 0 Type I Error Type II Error Correct results Correct results Truth Test result  1 -   1 -  H 0 True = statistically insignificant H 0 False = statistically significant Accept H 0 = statistically insignificant Reject H 0 = statistically significant Differences

33. Standard 2 X 2 table a = subjects with both the risk factor and the disease b = subjects with the risk factor but not the disease c = subjects with the disease but not the risk factor d = subjects with neither the risk factor nor the disease a + b = all subjects with the risk factor c + d = all subjects without the risk factor a + c = all subjects with the disease b + d = all subjects without the disease a + b + c + d = all study subjects PresentAbsent Present Absent Disease status Risk Factor Status ab cd a + b c + d a + c b + d a+b+c+d Total

34. Standard 2 X 2 table Sensitivity = a/a+c Specificity = d/b+d PresentAbsent Present Absent Disease status Risk Factor Status ab cd a + b c + d a + c b + d a+b+c+d Total

35. Diabetic Screening Program Sensitivity = a/a+c = 100 X 5/6 = 83.3% (16.7% false neg.) Specificity = d/b+d = 100 X 81/94 = 86.2%(13.8% false pos.) DiabeticNondiabetic >125mg/100ml <125mg/100ml Disease status Risk Factor Status 513 181 18 82 694100 Total