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. 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?
7. t-test for matched pairs
H0 there is no difference in the number of minutae when using ninhydrin
HA there are more minutae observed after the enhancement of ninhydrin.
1. Set up the null and alternative hypothesis:
8. t-test for matched pairs
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
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 0 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 > tcritical (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 > tcritical (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 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.]
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 = 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
61. t-test independent samples
.
Group A
(leading questions)
Group B
(control)
2 3
3 5
4 0
4 3
2 4
4 0
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 0
4 3
2 4
4 0
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
