Student t-test

Studnet's t-test for

1. Student t -test Unit 8 strand

2. Student’s t-test “Student” was W. S. Gossett . He published his test anonymously as ‘Student’ because he was working for the brewer’s Guinness and had to k eep the fact they were suing statistics a secret

3. Student’s t-test The test is used to compare samples from two different batches. It is usually used with small (<30) samples that are normally distributed .

4. There are two types of t-test: Matched pairs and independent pairs If there is some link between the data then use the matched pairs test ( eg a ‘before’ and ‘after’) If there is no link between the data then use the independent pairs test

5. DFO 8 12 11 6 9 11 7 8 10 9 DFO + ninhydrin 10 15 12 6 13 14 9 9 15 12 Question In an investigation to determine the effectiveness of sequencing of fingerprints 10 prints are taken enhanced with DFO and then with ninhydrin . The points of detail at each stage are recorded. Is there a difference at the 95% confidence level?

6. t-test for matched pairs 1. Set up the null and alternative hypothesis:

7. t-test for matched pairs H there is no difference in the number of minutae when using ninhydrin H A there are more minutae observed after the enhancement of ninhydrin . 1. Set up the null and alternative hypothesis:

8. t-test for matched pairs H there is no difference in the number of minutae when using ninhydrin H A there are more minutae observed after the enhancement of ninhydrin . This is a one-tail test. We are testing at the 95% or 5% (0.05) level 1. Set up the null and alternative hypothesis:

9. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample

10. t-test for matched pairs 1. Set up the null and alternative hypothesis DFO 8 12 11 6 9 11 7 8 10 9 DFO+ninhydrin 10 15 12 6 13 14 9 9 15 12 Difference (D) 2. Calculate the difference between the pairs in the sample

11. t-test for matched pairs 1. Set up the null and alternative hypothesis DFO 8 12 11 6 9 11 7 8 10 9 Total DFO+ninhydrin 10 15 12 6 13 14 9 9 15 12 Difference (D) 2 3 1 4 3 2 1 5 3 24 2. Calculate the difference between the pairs in the sample

12. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences

13. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences = 2.4

14. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences = 2.4

15. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference

16. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference

17. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference = 1.51

18. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error

19. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error

20. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error = = 0.478

21. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t

22. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t

23. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t = = 5.0

24. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and find the critical value

25. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and find the critical value DoF = No of pairs of data – 1

26. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and find the critical value DoF = No of pairs of data – 1 10-1 = 9 = n – 1

27. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

28. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

29. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

30. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

31. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

32. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 9. Determine if there is a difference or not 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

33. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 9. Determine if there is a difference or not 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

34. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 9. Determine if there is a difference or not t > t critical (5.0 > 1.833) 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

35. t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample 3. Calculate the mean of the differences 4. Calculate the standard deviation of the difference 5. Calculate the standard error 6. Calculate the value of t 9. Determine if there is a difference or not t > t critical (5.0 > 1.833) So, the null hypothesis is rejected and the alternative hypothesis is accepted. The ninhydrin does make a positive difference 7. Calculate the number of degrees of freedom ( DoF ) and 8. find the critical value

36. t-test independent samples If there is no before and after relationship between the samples then the independent samples test is used.

37. t-test independent samples If there is no before and after relationship between the samples then the independent samples test is used.

38. t-test independent samples Example Some brown dog hairs were found on the clothing of a victim at a crime scene involving a dog. The five of the hairs were measured: 46, 57, 54, 51, 38 μ m . A suspect is the owner of a dog with similar brown hairs. A sample of the hairs has been taken and their widths measured: 31, 35, 50, 35, 36 μ m . Is it possible that the hairs found on the victim were left by the suspect’s dog? Test at the %5 level. [From D. Lucy Introduction to Statistics for Forensic Scientists Chichester: Wiley, 2005 p. 44.]

39. t-test independent samples 1. Calculate the mean and standard deviation for the data sets

40. t-test independent samples Dog A Dog B 46 31 57 35 54 50 51 35 38 36 Total Mean Standard deviation 1. Calculate the mean and standard deviation for the data sets

41. t-test independent samples Dog A Dog B 46 31 57 35 54 50 51 35 38 36 Total 246 187 Mean 49.2 37.4 Standard deviation 7.463 7.301 1. Calculate the mean and standard deviation for the data sets

42. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 49.2 – 37.4 = 11.8 .

43. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference .

44. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means. │ 3. Calculate the standard error in the difference .

45. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . = = 4.669 ≈ 4.67 (3 sf )

46. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t

47. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . t = difference between the means ÷ standard error in the difference 4. Calculate the value of t :

48. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . t = difference between the means ÷ standard error in the difference 4. Calculate the value of t : 4.669 = 2.527 ≈ 2.53 (3 sig fig) 11.8

49. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom

50. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom = n 1 + n 2 - 2

51. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom = n 1 + n 2 - 2 5 + 5 - 2 = 8

52. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to from the table

53. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to from the table

54. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to and find the critical value from the table

55. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to and find the critical value from the table

56. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to from the table At the 0.05 level t crit = 2.306

57. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to and find the critical value from the table If t < critical value then there is no significant difference between the two sets of data If t > critical value then there is a significant difference between the two sets of data

58. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to and find the critical value from the table If t < critical value then there is no significant difference between the two sets of data If t > critical value then there is a significant difference between the two sets of data So, at 0.05 level there is a significant difference between the two data sets. So it could not come from the same dog.

59. t-test independent samples 1. Calculate the mean and standard deviation for the data sets 2. Calculate the magnitude of the difference between the two means 3. Calculate the standard error in the difference . 4. Calculate the value of t : 5. Calculate the degrees of freedom 6. Find the critical value for the particular significance you are working to and find the critical value from the table

60. t-test independent samples . Group A (leading questions) Group B (control) 2 3 3 5 4 4 3 2 4 4 3

61. t-test independent samples . Group A (leading questions) Group B (control) 2 3 3 5 4 4 3 2 4 4 3 Mean = 3.143 SD= 0.8997 Mean = 2.50 SD = 2.07

62. t-test independent samples . Group A (leading questions) Group B (control) 2 3 3 5 4 4 3 2 4 4 3 Mean = 3.143 SD= 0.8997 Mean = 2.50 SD = 2.07 Diff of means = 0.64 SE of diff = 0.861 DoF = 11 t = 0.7433

63. t-test independent samples . DoF = 11 t = 0.7433

64. t-test independent samples . DoF = 11 t = 0.7433

65. t-test independent samples . DoF = 11 t = 0.7433 t < t critical at 5% level, 2-tail There is no significant difference between the data test