t test and t distribution
 t test is alternative to Z test when population standard
deviation is unknown
 There is family of t distributions rather than single
distributions
 In general t distribution is flatter than z distribution

t test
• For single sample
• For independent samples design
• For repeated measures design

t test for single sample
Assumptions
1. Random sampling
2. Independent observations
3. Normal sampling distribution

• t =
• Whereas sM =
• s2 =
• SS =
M
s
X 

n
s2
1

n
SS
   2
2
2


 
 X
X
or
n
X
X

• Problem
A survey in a school district revealed that the
average age at which students first tried an
alcoholic drink was 14 years. To determine
whether anything has changed, a random
sample of students was asked questions about
alcohol use. The age at which drinking behavior
first began was reported by members of the
sample as follows: 11,13, 4,12, 10. Has there
been a change in the mean age at which
drinking began?

Effect Size
Cohen’s d =
s
X 


Independent measures t test
Assumptions
Observations must be independent within each
sample and between samples
Normal sampling distributions of the two
populations from which the samples were
selected
The two populations must have equal
variances

 
2
1
2
1
X
X
s
X
X
t



2
1
2
1
2
2
2
1
2
2
1
df
df
SS
SS
s
and
n
s
n
s
s
Whereas p
P
P
X
X







• In a study of jury behavior, two samples of subjects were
provided details about a trial in which the defendant was
obviously guilty. Although group 2 received the same
details as group 1, the second group was also told that
some evidence had been withheld from the jury by the
judge. Later the subjects were asked to recommend a
jail sentence. The length of term suggested by each
subject is presented here. Is there a significant
difference between the two groups in their responses,
supposing that literature review suggests that length of
the term suggested by group 2 should be more than as
suggested by group 1

Data
Group 1 Group2
6 11
5 12
4 12
4 10
5 12
5 12
4 13
3 12

Effect Size
Cohen’s d =
sp
X
X 2
1 

t test for repeated
measures designs

USe
• For the comparison of DV at two different
times in the same sample
• For the comparison of two variables in the
same sample

Assumptions
• The observations within each treatment
must be independent
• The sampling distribution of difference
scores (D values) must be normal

Formula
n
s
s
Whereas
s
D
t D
D
D
2



Problem
The investigator decides to examine the effect of
relaxation training on the severity of asthma
symptoms. A sample of five patients is selected for
the study. During the week before treatment, the
investigator records the severity of symptoms by
measuring how many doses of medication are
needed for asthma attacks. Then the patients
receive relaxation training. For the week following
training, the researcher once again records the
number of doses required by each patient. Do the
data indicate that relaxation training alters the
severity of symptoms?

Data
Subjects Before treatment After treatment
A 9 4
B 4 1
C 5 5
D 4 0
E 5 1

?
• What is the independent variable?
• What is the dependent variable?
• What is the degree of freedom?
• Operationally define the dependent
variable?
• What are the scores in the data?

Effect Size
D
s
D
d
s
Cohen 
'