1. Chapter 11 The t-Test for Two Related Samples
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J Gravetter and Larry B. Wallnau

2. Chapter 11 Learning Outcomes
Understand structure of research study appropriate for repeated-measures t hypothesis test 2
Test mean difference between two treatment conditions using repeated-measures t statistic 3
Evaluate effect size using Cohen’s d, r2, and/or a confidence interval 4
Explain pros and cons of repeated-measures and independent measures studies

3. Tools You Will Need Introduction to the t Statistic (Chapter 9)
Estimated standard error
Degrees of freedom
t Distribution
Hypothesis test with t statistic
Independent-Measures Design (Chapter 10)

4. 11.1 Introduction to Repeated-Measures Designs
Also known as within-subjects design
Two separate scores are obtained for each individual in the sample
Same subjects are used in both treatment conditions
No risk of the participants in each treatment group differing significantly from each other

5. Matched-Subjects Design
Approximates the advantages of a repeated-measures design
Two separate samples are used
Each individual in a sample is matched one-to-one with an individual in the other sample.
Matched on relevant variables
Participants are not identical to their match
Ensures that the samples are equivalent with respect to some specific variables

6. Related-Samples Designs
Related (or correlated) sample designs
Repeated-measures
Matched samples
Statistically equivalent methods
Use different number of subjects
Matched sample has twice as many subjects as a repeated-measures design

7. 11.2 t Statistic for Repeated- Measures Research Design
Structurally similar to the other t statistics
Essentially the same as the single-sample t
Based on difference scores (D) rather than raw scores (X)
Difference score = D = X2—X1
Mean Difference

8. Hypotheses for Related-Samples t Test
H0: μD = 0
H1: μD ≠ 0

9. Figure 11.1 Populations of Difference Scores
FIGURE (a) A population of difference scores for which the mean is μD = 0. Note that the typical difference score (D value) is not equal to zero. (b) A population of difference scores for which the mean is greater than zero. Note that most of the difference scores are also greater than zero.

10. t- Statistic for Related Samples

11. Figure 11.2 Difference Scores for 4 People Measured Twice
FIGURE A sample of n = 4 people is selected from the population. Each individual is measured twice, once in treatment I and once in treatment II, and a difference score, D, is computed for each individual. This sample of difference scores is intended to represent the population. Note that we are using a sample of difference scores to represent a population of difference scores. Note also that the mean for the population of difference scores is unknown. The null hypothesis states that, for the general population, there is no consistent or systematic difference between the two treatments, so the population mean difference is μD = 0.

12. Learning Check
For which of the following would a repeated-measures study be appropriate? A matched-subjects study? A
A group of twins is tested for IQ B
Comparing boys and girls strength at age 3 C
Evaluating the difference in self-esteem between athletes and non-athletes D
Students’ knowledge is tested in September and December

13. Learning Check - Answer
For which of the following would a repeated-measures study be appropriate? A matched-subjects study? A
A group of twins is tested for IQ (matched) B
Comparing boys and girls strength at age 3 C
Evaluating the difference in self-esteem between athletes and non-athletes D
Students’ knowledge is tested in September and December (repeated-measures)

14. Learning Check
Decide if each of the following statements is True or False
T/F
A matched-samples study requires only 20 participants to obtain 20 scores in each of the conditions being compared
As the variance of the difference scores increases, the magnitude of the t statistic decreases

15. Learning Check - Answers
False
Matched sample would require 20 subjects matched to 20 additional subjects
True
Increasing the variance increases the denominator and decreases the t statistic

16. 11.3 Repeated-Measures Design Hypothesis Tests and Effect Size
Numerator of t statistic measures actual difference between the data MD and the hypothesis μD
Denominator measures the standard difference that is expected if H0 is true
Same four-step process as other tests

17. Figure 11.3 Critical region for t df = 8 and α = .05
FIGURE The critical region for the t distribution with df = 8 and α = .05.

18. Effect size for Related Samples

19. In The Literature
Report means and standard deviation in a statement or table
Report a concise version of test results
Report t values with df
Report significance level
Report effect size
E.g., t(9) = 2.43, p<.05, r2 = .697

20. Factors That Influence Hypothesis Test Outcome
Size of the sample mean difference (larger mean difference  larger numerator so increases t
Sample size (larger sample size  smaller standard error—denominator—so larger t)
Larger sample variance  larger standard error—denominator—so larger t)

21. Variability as measure of consistency
When treatment has consistent effect
Difference scores cluster together
Variability is low
When treatment effect is inconsistent
Difference scores are more scattered
Variability is high
Treatment effect may be significant when variability is low, but not significant when variability is high

22. Figure 11.4 Example 11.1 Consistent Difference Scores
FIGURE The sample of difference scores from Example The mean is MD = -2 and the standard deviation is s = 2. The difference scores are consistently negative, indicating a decrease in perceived pain, suggesting that μD = 0 (no effect) is not a reasonable hypothesis.

23. Figure 11.5 A Sample of Inconsistent Difference Scores
FIGURE A sample of difference scores with a mean of MD = -2 and a standard deviation of s = 6. The data do not show a consistent increase or decrease in scores. Because there is no consistent treatment effect, μD = 0 is a reasonable hypothesis.

24. Directional Hypotheses and One-Tailed Tests
Researchers often have specific predictions for related-samples designs
Null hypothesis and research hypothesis are stated directionally, e.g.
H0: μD ≤ 0
H1: μD > 0
Critical region is located in one tail

25. 11.4 Related-Samples Vs. Independent-Samples t Tests
Advantages of repeated-measures design
Requires fewer subjects
Able to study changes over time
Reduces or eliminates influence of individual differences
Substantially less variability in scores

26. 11.4 Related-Samples Vs. Independent-Samples t Tests
Disadvantages of repeated-measures design
Factors besides treatment may cause subject’s score to change during the time between measurements
Participation in first treatment may influence score in the second treatment (order effects)
Counterbalancing is a way to control time-related or order effects

27. Related-Samples t Test Assumptions
Observations within each treatment condition must be independent
Population distribution of difference scores (D values) must be normally distributed
This assumption is not typically a serious concern unless the sample size is small.
With relatively large samples (n > 30) this assumption can be ignored

28. Learning Check n = 15 and SS = 10 n = 15 and SS = 100
Assuming that the sample mean difference remains the same, which of the following sets of data is most likely to produce a significant t statistic? A
n = 15 and SS = 10 B
n = 15 and SS = 100 C
n = 30 and SS = 10 D
n = 30 and SS = 100

29. Learning Check - Answer
Assuming that the sample mean difference remains the same, which of the following sets of data is most likely to produce a significant t statistic? A
n = 15 and SS = 10 B
n = 15 and SS = 100 C
n = 30 and SS = 10 D
n = 30 and SS = 100

30. Learning Check
Decide if each of the following statements is True or False
T/F
Compared to independent-measures designs, repeated-measures studies reduce the variance by removing individual differences
The repeated-measures t statistic can be used with either a repeated-measures or a matched-subjects design

31. Learning Check - Answers
True
Using the same subjects in both treatments removes individual differences across treatments
Both of these related-samples tests reduce individual differences across treatments

32. Figure 11.6 Example 11.1 SPSS Repeated-Measures Test Output
FIGURE The SPSS output for the repeated-measures hypothesis test in Example 11.1.

33. Any Questions?
Concepts?
Equations?