The document describes how to perform a student's t-test to compare two samples. It provides steps for both a matched pairs t-test and an independent samples t-test. For a matched pairs t-test, the steps are: 1) state the null and alternative hypotheses, 2) calculate the differences between pairs, 3) calculate the mean difference, 4) calculate the standard deviation of the differences, 5) calculate the standard error, 6) calculate the t value, 7) determine the degrees of freedom, 8) find the critical t value, and 9) determine if there is a statistically significant difference. For an independent samples t-test, similar steps are followed to calculate means, standard deviations, the difference between

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 keep
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. 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?
DFO 8 12 11 6 9 11 7 8 10 9
DFO+ 10 15 12 6 13 14 9 9 15 12
ninhydrin

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

7. t-test for matched pairs
1. Set up the null and alternative hypothesis:
H0 there is no difference in the number of minutae when using ninhydrin
HA there are more minutae observed after the enhancement of ninhydrin.

8. t-test for matched pairs
1. Set up the null and alternative hypothesis:
H0 there is no difference in the number of minutae when using ninhydrin
HA 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

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
2. Calculate the difference between the pairs in the sample
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)

11. t-test for matched pairs
1. Set up the null and alternative hypothesis
2. Calculate the difference between the pairs in the sample
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 0 4 3 2 1 5 3 24

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
=n–1
10-1 = 9

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
7. Calculate the number of degrees of freedom (DoF) and 8. find the critical value
9. Determine if there is a difference or not

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
7. Calculate the number of degrees of freedom (DoF) and 8. find the critical value
9. Determine if there is a difference or not

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
7. Calculate the number of degrees of freedom (DoF) and 8. find the critical value
9. Determine if there is a difference or not
t > tcritical (5.0 > 1.833)

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
7. Calculate the number of degrees of freedom (DoF) and 8. find the critical value
9. Determine if there is a difference or not
t > tcritical (5.0 > 1.833)
So, the null hypothesis is rejected and the alternative
hypothesis is accepted.
The ninhydrin does make a positive difference

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 μm.
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 μm.
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
1. Calculate the mean and standard deviation for the data sets
Dog A Dog B
46 31
57 35
54 50
51 35
38 36
Total
Mean
Standard
deviation

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

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
4. Calculate the value of t:
t = difference between the means ÷ standard error in the difference

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
4. Calculate the value of t:
t = difference between the means ÷ standard error in the difference
11.8 4.669 = 2.527
≈ 2.53 (3 sig fig)

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 = n1 + n2 - 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 = n1 + n2 - 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 tcrit = 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 Group B
.
(leading questions) (control)
2 3
3 5
4 0
4 3
2 4
4 0
3

61. t-test independent samples
Group A Group B
.
(leading questions) (control)
2 3
3 5
4 0
4 3
2 4
4 0
3
Mean = 3.143 Mean = 2.50
SD= 0.8997 SD = 2.07

62. t-test independent samples
Group A Group B
.
(leading questions) (control)
2 3
3 5
4 0
4 3
2 4
Diff of means
4 0 = 0.64
3 SE of diff = 0.861
Mean = 3.143 Mean = 2.50 DoF = 11
SD= 0.8997 SD = 2.07 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 < tcritical at 5% level, 2-tail
There is no significant difference between the data test