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The t Test for Two Related Samples
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 test1
• Test mean difference between two treatment
conditions using repeated-measures t statistic2
• Evaluate effect size using Cohen’s d, r2, and/or a
confidence interval3
• Explain pros and cons of repeated-measures and
independent measures studies4
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
• Repeated-measures design
– 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
n
D
MD
12. Learning Check
• For which of the following would a repeated-
measures study be appropriate?
A matched-subjects study?
• A group of twins is tested for IQA
• Comparing boys and girls strength at age 3B
• Evaluating the difference in self-esteem
between athletes and non-athletesC
• Students’ knowledge is tested in September
and December
D
13. Learning Check - Answer
• For which of the following would a repeated-
measures study be appropriate?
A matched-subjects study?
• A group of twins is tested for IQ (matched)A
• Comparing boys and girls strength at age 3B
• Evaluating the difference in self-esteem
between athletes and non-athletesC
• Students’ knowledge is tested in September
and December (repeated-measures)D
14. Learning Check
• Decide if each of the following statements
is True or False
• A matched-samples study requires only
20 participants to obtain 20 scores in
each of the conditions being compared
T/F
• As the variance of the difference scores
increases, the magnitude of the t
statistic decreases
T/F
15. Learning Check - Answers
• Matched sample would require 20
subjects matched to 20 additional
subjects
False
• Increasing the variance increases
the denominator and decreases
the t statistic
True
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
18. Effect size for Related Samples
s
M
dsCohen'estimated D
dft
t
r
2
2
2
DMDD stMIC :)1.(.
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
23. Figure 11.5 A Sample of
Inconsistent Difference Scores
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
• 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?
• n = 15 and SS = 10A
• n = 15 and SS = 100B
• n = 30 and SS = 10C
• n = 30 and SS = 100D
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?
• n = 15 and SS = 10A
• n = 15 and SS = 100B
• n = 30 and SS = 10C
• n = 30 and SS = 100D
30. Learning Check
• Decide if each of the following statements
is True or False
• Compared to independent-measures
designs, repeated-measures studies reduce
the variance by removing individual
differences
T/F
• The repeated-measures t statistic can be
used with either a repeated-measures or a
matched-subjects design
T/F
31. Learning Check - Answers
• Using the same subjects in both
treatments removes individual
differences across treatments
True
• Both of these related-samples
tests reduce individual differences
across treatments
True
FIGURE 11.1 (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.
FIGURE 11.2 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.
FIGURE 11.3 The critical region for the t distribution with df = 8 and α = .05.
FIGURE 11.4 The sample of difference scores from Example 11.1. 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.
FIGURE 11.5 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.
FIGURE 11.6 The SPSS output for the repeated-measures hypothesis test in Example 11.1.