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ttest for Correlated Samples

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Title: ttest for Correlated Samples


1
t-test for Correlated Samples
  • A researcher believes that relaxation training
    will reduce the severity of asthma attacks in
    asthmatic subjects. A sample of five subjects is
    chosen for the study. The researcher records the
    severity of their symptoms by measuring how many
    doses of medication are needed for asthma
    attacks. The subjects then receive relaxation
    training. The week after, the researcher once
    again measures the number of doses required by
    each patient. The data are provided below. Is
    the researcher correct? Use normal decision
    rules?
  • This is a Before and After design.

2
D 5 3 0 4 4
D2
25 9 0 16 16
?D2 66
?D 16
Note, D is the difference between each patients
score before and after training.
3
  • H0 ?D 0 (No change in symptoms)
  • H1 ?D gt 0 (There is a change)

Calculate the sums of squares of the difference
scores.
N number of subjects per group
66 - (16)2 5
14.8
4
Calculate the standard error of the difference
between the means.
0.86
Now we can calculate t.
t 3.2 - 0 3.72 0.86
Calculate df
df N - 1 Where N number of
subjects per group
5 - 1 4
t crit 4 2.132
5
  • Since t obs 3.72 gt t crit4 2.132 we
    reject the H0. There is a change in symptoms
    following relaxation training (t obs 3.72, p lt
    0.05).

6
t-test for Correlated Samples
  • A researcher wishes to compare a new method of
    teaching reading to slow learners to the
    current standard method. Eight pairs of slow
    learners with similar IQs are found, and one
    member of each pair is randomly assigned to the
    standard teaching method, while the other is
    assigned to the new method. Reading ability is
    then measured with a standard reading test. Is
    one method better than the other? Use normal
    decision rules.
  • This is a matched group design because subjects
    are paired based on their IQ scores.

7
Pair New Standard
Difference Method
Method 1 77 72 5 2 74 68 6 3
82 76 6 4 73 68 5 5 87 84 3
6 69 68 1 7 66 61 5 8
80 76 4
D 4.375
?D 35
?D2 173
8
  • H0 µD 0
  • H1 µD 0

Calculate sums of squares.
173 (35)2 19.875 8
9
Calculate standard error
Calculate t
4.375 7.343 0.5957
Calculate df
df N - 1 Where N number of pairs
8 1 7
t crit 2.365
Since tobs 7.343 gt t crit7 2.365, we
reject the H0. The new method leads to
significantly higher reading scores (t7 7, p lt
0.05)
10
More Examples
  • E.g. An investigator wants to determine whether
    whether a banned performance enhancing drug
    increases endurance when injected into an athlete
    just before a competitive event. Ten volunteer
    athletes are chosen and randomly assigned into
    two groups of 5. The first group is given the
    banned substance whereas the second group is
    injected with a harmless red fluid. The subjects
    then run on a treadmill until exhaustion. Total
    time on treadmill is measured. Is there a
    difference between the two group? Use normal
    decision rules.
  • Independent t-test with equal N.

11
N 5 N 5 X1 7
X2 6 ?X1 35 ?X2 30
?X12 259 ?X22 190
12
H0 ?1 - ?2 0
H1 ?1 - ?2 0
259 - (35)2 5
190 - (30)2 5
14
10
13
1.10
t (X1 - X2) - (?1 - ?2) SX1 - X2
7 - 6 1.10
0.91
df N1 N2 - 2 5 5 - 2 8
t crit 8 2.306
Since t obs 0.91 lt t crit 2.306, do not
reject H0. There is no difference between the
groups (t 8 0.91, p gt 0.05)
14
More Examples
  • A researcher wishes to investigate the claim that
    vitamin C reduces the frequency of the common
    cold. To eliminate the variability due to
    different family environments, pairs of children
    from the same family are randomly assigned to
    either a group that receives vitamin C or a group
    that receives fake vitamin C. The researcher
    then measures number of days sick during the
    school year. Is there a difference in frequency
    of sickness between the two groups? Use normal
    decision rules.
  • Matched group design.

15
Days Missed Due to Illness Pair No.
Vitamin C Fake Vitamin C 1
2 3 2 5 4 3 7 9 4 0 3 5 3 5 6
7 7 7 4 6 8 5 8 9 1 2 10 3 5
D -1 1 -2 -3 -2 0 -2 -3 -1 -2
D -1.5 ?D -15 ?D2 37
16
H0 ? D 0
H1 ? D lt 0
SSD ?D2 - (?D)2 N
37 - (-15)2 14.5 10
0.401
-1.5 - 0 -3.74 0.401
df N - 1 10 - 1 9
t crit 9 1.833
Since t obs -3.74 gt t crit -1.833, reject
H0. Children who take vitamin C have
significantly fewer illnesses (t9 -3.74, p lt
0.05)
17
More Examples
  • A researcher believes that women show more
    empathy than men. Five men and four women are
    interviewed. The number of times in which they
    show empathy is recorded. Do the results support
    the researchers hypothesis? Use normal decision
    rules.
  • Independent t-test for unequal N.

18
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19
H0 ?1 - ?2 0
H1 ?1 - ?2 lt 0
262 - (32)2 4
100 - (20)2 5
20
6
20
1.29
21
4 - 8 -3.10 1.29
df N1 N2 - 2 5 4 - 2 7
t crit 7 -1.895
Since t obs -3.10 gt t crit -1.895, reject
H0. Women show significantly more empathy than
men (t 7 -3.10, p lt 0.5)
22
More Examples
  • A researcher is interested in attitudes towards
    public schools. A group of ten subjects is given
    a test (pretest) regarding their attitudes and
    then shown a movie about public schools.
    Following the movie, the group is tested again
    (posttest). Are attitudes as measured by the
    tests were more or less favorable after seeing
    the movie than they were before? Use normal
    decision rules.
  • This is a before and after design.

23
D 5 ?D 50 ?D2 322
24
H0 ?D 0
H1 ?D 0
SSD ?D2 - (?D)2 N
322 - (50)2 72 10
0.894
5 - 0 5.59 0.894
25
  • df N - 1 10 - 1 9
  • t crit 9 2.262
  • Since t obs 5.59 gt t crit 2.262, we reject
    H0. Attitudes after the movie are different from
    attitudes before the movie (t 9 5.59, p lt 0.05).
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