1. Chapter 10: The t Test For Two Independent Samples

2. Independent-Measures Designs
The independent-measures hypothesis test allows researchers to evaluate the mean difference between two populations using the data from two separate samples.
The identifying characteristic of the independent-measures or between-subjects design is the existence of two separate or independent samples.
Thus, an independent-measures design can be used to test for mean differences between two distinct populations (such as men versus women) or between two different treatment conditions (such as drug versus no-drug).

3. Population A vs. Population B
µ = 450
 = 100
µ = ?
 = 100 z

4. Z-formula broken down Population Sample Statistic Parameter
Standard Error
(Sampling Error)

5. Population A vs. Population B based on sample data
µ = 30
 = ?
µ = 30
 = ? t

6. Estimated Standard Error
t-formula broken down
Population
Parameter
Sample Statistic
Estimated Standard Error
(Sampling Error)

7. Different populations using different teaching methods
Population A
Taught by method A
Population B
Taught by method B
Unknown
µ = ?
Unknown
µ = ?
Sample A
Sample B

8. Goal: Evaluate Mean difference Between 2 Population Means
Ho: µ1 - µ2 = 0 (No difference between the population means)
H1: µ1 - µ2 ≠ 0 (There is a difference between the pop means)

9. Procedure for obtaining empirical sampling distribution of differences between two means
Population 1
too large to measure in its entirety;
in fact:
Population 2
too large to measure in its entirety;
in fact:
Normal
Normal
12 = 22
22 = 12
µ1 = 80
µ2 = 80
Difference
(X1 - X2)
( ≠ 0 due to
sampling error)
Random
Random
Sample 1
N1 = 30
X1 = 81.6
Sample 2
N1 = 30
X1 = 79.7
Trial 1
+1.9
Sample 1
N1 = 30
X1 = 78.2
Sample 2
N1 = 30
X1 = 79.4
Trial 2
-1.2
Sample 1
N1 = 30
X1 = 80.1
Sample 2
N1 = 30
X1 = 79.5
Trial 1000
+0.6

10. Table of Mean Differences

11. Frequency polygon for mean differences table
150
100
Frequency (f) 50
-10 -5 5 10
Difference between sample means (X1 - X2)

12. t-formula for comparing sample means
Sample Statistic
Population
Parameter
Estimated Standard Error
(Sampling Error)

13. t-formula (for sample means) broken down
Sample Statistic
Population
Parameter
Sampling Error
(Standard Error of the Mean Difference)

14. Standard Error of the Means formula

15. Population II vs. Population I10 20 30 40 50 60 70 80

16. Pooled Variance Formulas

17. Final Formula for Independent t
df = df 1st sample + df 2nd sample
= df1 + df2
OR df = (n1 - 1) + (n2 - 1)

18. 5 step process for t statistic
1. Ho: µ1 - µ2 = 0 (No effect of…)
H1: µ1 - µ2 ≠ 0 (Effect…) 
2. Critical region for t statistic
df = df1 + df2 p < 
tcritical =
 =
3. Collect data and compute test statistic
4. Make a decision
5. Conclusion

19. Do mental images help memory?
40 pairs of nouns
(dog / bicycle, lamp / piano, etc.)
2 groups (or samples)
1. Memorization (control)
2. Imagery (experimental)
Subsequent memory test

20. Data (Number of words recalled)
Example p.274
Data (Number of words recalled)
Group 1
Group 2
(No images)
(Images) 24 13 18 31 23 17 19 29 16 20 26 15 21 30

21. Data (Number of words recalled)
Example p.274
Data (Number of words recalled)
Group 1
Group 2
(No images)
(Images) 24 13 18 31 23 17 19 29 16 20 26 15 21 30
n = 10
n = 10
X = 19
X = 25
SS = 160
SS = 200

22. Critical regions
Reject Ho
Reject Ho
Fail to reject Ho
-2.101
2.101

23. t formula and standard error formula

24. Table demonstrating long Sum of Squares computationX1
(X1 - X1)
(X1 - X1)2 24 19 +5 25 23 +4 16 -3 9 17 -2 4 13 -6 36 20 +1 1 15 -4 26 +7 49

25. Table demonstrating short SS computationX1
X12 24
576 23
529 16
256 17
289 19
361 13
169 20
400 15
225 26
676
∑x = 190
∑x2 = 3770

26. Figure p. 276
df = 18
Reject H0
-1.734

27. Assumptions Underlying the Independent-Measures t statistic
The observation within each sample must be independent
The two populations from which the samples come must be normal
The two populations from which the samples are selected must have equal variances - homogeneity of variance

28. Homogeneity Assumption: How do we know when we have violated it?
For small samples (n < 10):
If one of the sample variances is more than 4 times larger than the other
For larger samples:
If one of the sample variances is more than 2 times larger than the other

29. Frequency Histogram of Attitude Scores: SS = 4
Women
Men
n = 7
n = 7
X = 8.00
X = 12.00
SS = 4
SS = 4 4 3
Frequency 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Attitude Scores

30. Frequency Histogram of Attitude Scores: SS = 242
Women
Men
n = 7
n = 7
X = 8.00
X = 12.00
SS = 242
SS = 242 4 3
Frequency 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Attitude Scores