The t Test for Two Related Samples

THE t TEST FOR TWO
RELATED SAMPLES
Repeated-Measures Design
Repeated Measures Design
• Repeated-measures design/ within-subject design is one in which a single sample of
individuals is measured more than once on the same dependent variable.
• Same subjects are used in all of the treatment conditions.
• Advantage – there is no risk that participants in one treatment group are different from
the participants in the other group as you are using the same individuals
• An alternative to is to select groups using ‘matched subjects’ design: each individual in
one sample is matched with an individual in the other sample i.e. the two individuals
are from similar ethnic backgrounds, similar ages, same sex etc.
When do we use a repeated-measures
design?
Example:
You want to test if eating honey before training makes individuals run faster. You take your sample and record their
running time before you give them honey. Then you give them honey and record their running time again.
Step 1 : Difference score (D score)= X2-X1
Step 2: Calculate MD for the difference scores and SS
Step 3: H0: μD=0 and H1: μD≠0
MD−μD
Step 4:
t=
Step 5:
s2=𝑛−1
s=
Step 6:
𝑆𝑀𝐷 =
𝑠2 𝑠
=
𝑛
𝑛
𝑆 𝑀𝐷
𝑆𝑆
𝑆𝑆
𝑑𝑓
More on related-samples t test
Practices:
Factors other than the treatment effect could cause participant’s score to change, not the
treatment such as the passage of time (time effect). Also, as you are measuring at two different
time points, individuals mood, health might change (order effect)
To deal with this problem time-related factors or order effects is to counterbalance the
presentation of treatments: Divide individuals in to two groups, give one group treatment 1
followed by treatment 2, and the second group treatment 2 and than 1 OR use matchedsubjects design.
Assumptions:
1.
The observations within each treatment condition must be independent. Scores within
each treatment is independent.
2.
The population distribution of difference scores must be normal (if n>30, this assumption
can be ignored)
Questions
A researcher uses a repeated-measures study to compare two treatment conditions with
a set of 20 scores in each treatment. What would be the value of df for the repeatedmeasures t statistic?
a. df = 19
b. df = 38
c. df = 18
d. df = 39
Which of the following is the correct null hypothesis for a repeated-measures t test?
a. µ1 = µ2
b. MD = 0
c. µD = 0
d. M1 = M2
For a repeated-measures study comparing two treatments with a sample of n = 9
participants, the difference scores have a mean of M = 4.90 with SS = 288. What is the
estimated standard error for the sample mean difference?
a. 2
b. 36/8 = 4.5
c. 4
d. 36
A researcher obtains a t statistic of t = 2.00 from a repeated measures study using n = 7
participants. If the effect size is measured using r2 then the value of r2 for the study would
be ________.
a. r2 = 4/10 = 0.40
b. r2 = 4.00
c. r2 = 4/40 = 0.10
d. The value of r2 cannot be determined from the information provided.
A researcher obtains t = 4.00 and MD = 9 for a repeated-measures study. If the
researcher measures effect size using the percentage of variance accounted for, what
value will be obtained for r2?
a. Cannot determine without additional information.
b. 16/81
c. 16/97
d. 9/4
A researcher obtains t = 2.35 for a repeated-measures study using a sample of n = 8
participants. Based on this t value, what is the correct decision for a two-tailed test?
a. Reject the null hypothesis with either α= .05 or α = .01
b. Reject the null hypothesis with α = .05 but not with α = .01
c. Cannot make a decision without additional information.
d. Fail to reject the null hypothesis with either α = .05 or α = .01
A research report describing the results from a repeated-measures study states: The
data show no significant difference between the two treatments, t(10) = 1.65, p > .05.
Based on this report, you can conclude that a total of ________ individuals participated in
the research study.
a. 10
b. 9
c. 11
d. 12
In general, if the variance of the difference scores increases, then what will happen to the
value of the t statistic?
a. It will stay the same - the t statistic is not affected by the variance of the difference
scores.
b. It will decrease (move toward 0 at the center of the distribution).
c. It may increase or may decrease. There is no consistent relationship between
variance and the size of the t statistic.
d. It will increase (move farther toward the tail of the distribution).
Which of the following samples will produce the largest value for a t statistic? Assume
each sample has n = 10 scores.
a. MD = 10 with SS = 20
b. MD = 5 with SS = 20
c. MD = 10 with SS = 40
d. MD = 5 with SS = 40
Compared to an independent-measures design, a repeated-measured study is more
likely to find a significant effect because it reduces the contribution of variance due to
________.
a. MD
b. degrees of freedom
c. the effect of the treatment
d. individual differences
A researcher would like to examine how the chemical tryptophan, contained in foods such
as turkey, can affect mental alertness. A sample of n = 9 college students is obtained and
each student's performance on a familiar video game is measured before and after eating
a traditional Thanksgiving dinner including roast turkey. The average score dropped by
M = 14 points after the meal with SS = 1152 for the difference scores.
a. Is there is significant difference in performance before eating versus after eating? Use
a two-tailed test with = .05.
b. Compute r2 to measure the size of the effect.
c. Write a sentence demonstrating how the outcome of the test and the measure of effect
size would appear in a research report.
A researcher conducts an independent-measures study examining how the brain chemical
serotonin is related to aggression. One sample of rats serves as a control group and receives a
placebo that does not affect normal levels of serotonin. A second sample of rats receives a
drug that lowers brain levels of serotonin. Then the researcher tests the animals by recording
the number of aggressive responses each of the rats display. The data are as follows.
a. Does the drug have a significant effect on aggression? Use an alpha level of .05, two tails.
b. Compute Cohen's d to measure the size of the treatment effect.