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.
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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
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QuestionIn an investigation to determine the effectiveness of sequencing of fingerprints10 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
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t-test for matched pairs 1. Set up the null and alternative hypothesis:
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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.
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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
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t-test for matched pairs 1. Set up the null and alternative hypothesis 2. Calculate the difference between the pairs in the sample
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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
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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
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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
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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
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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
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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.
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t-test independent samplesExampleSome brown dog hairs were found on the clothing of a victim at a crime sceneinvolving 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 hairshas 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.]
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t-test independent samples1. Calculate the mean and standard deviation for the data sets
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t-test independent samples1. 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
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t-test independent samples1. 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
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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
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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
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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
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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
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t-test independent samples. DoF = 11 t = 0.7433
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t-test independent samples. DoF = 11 t = 0.7433
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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
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