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Inference about Means and Mean Differences
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Chapter 9
INTRODUCTION TO
t STATISTIC
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Inference about Means and Mean Differences
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Preview-1
 In the previous chapter, we presented the
statistical procedure that permit researcher to
use sample mean to test hypothesis about an
unknown population
 Remember that the expected value of the
distribution of sample means is μ, the
population mean
σM =
© aSup-2007
σ
√n
M-μ
z= σ
M
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Inference about Means and Mean Differences
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THE PROBLEM WITH z-SCORE
 A z-score requires that we know the value of
the population standard deviation (or
variance), which is needed to compute the
standard error
 In most situation, however, the standard
deviation for the population is not known
 In this case, we cannot compute the standard
error and z-score for hypothesis test. We use t
statistic for hypothesis testing when the
population standard deviation is unknown
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Inference about Means and Mean Differences
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THE t STATISTIC:
AN ALTERNATIVE TO z
The goal of the hypothesis test is to
determine whether or not the obtained
result is significantly greater than would be
expected by chance.
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Inference about Means and Mean Differences
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Introducing t Statistic
Now we will estimates the standard
error by simply substituting the
sample variance or standard
deviation in place of the unknown
population value
SM =
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s
√n
σM =
σ
√n
Notice that the symbol for estimated
standard error of M is SM instead of
σM , indicating that the estimated
value is computed from sample data
rather than from the actual population
parameter
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Inference about Means and Mean Differences
z-score and t statistic
σM =
σ
√n
M-μ
z= σ
M
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SM =
t=
s
√n
M-μ
SM
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Inference about Means and Mean Differences
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The t Distribution
 Every sample from a population can be
used to compute a z-score or a statistic
 If you select all possible samples of a
particular size (n), then the entire set of
resulting z-scores will form a z-score
distribution
 In the same way, the set of all possible t
statistic will form a t distribution
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Inference about Means and Mean Differences
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The Shape of the t Distribution
 The exact shape of a t distribution
changes with degree of freedom
 There is a different sampling distribution
of t (a distribution of all possible sample t
values) for each possible number of
degrees of freedom
 As df gets very large, then t distribution
gets closer in shape to a normal z-score
distribution
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Inference about Means and Mean Differences
HYPOTHESIS TESTS WITH t STATISTIC
 The goal is to use a sample from the
treated population (a treated sample) as
the determining whether or not the
treatment has any effect
Unknown population
after treatment
Known population
before treatment
TREATMENT
μ = 30
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μ=?
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Inference about Means and Mean Differences
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HYPOTHESIS TESTS WITH t STATISTIC
 As always, the null hypothesis states that the
treatment has no effect; specifically H0 states
that the population mean is unchanged
 The sample data provides a specific value for
the sample mean; the variance and estimated
standard error are computed
sample mean
population mean
(from data)
(hypothesized from H0)
t=
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-
Estimated standard error
(computed from the sample data)
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Inference about Means and Mean Differences
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LEARNING CHECK
A psychologist has prepared an “Optimism Test”
that is administered yearly to graduating college
seniors. The test measures how each graduating
class feels about it future. The higher the score, the
more optimistic the class. Last year’s class had a
mean score of μ = 19. A sample of n = 9 seniors from
this years class was selected and tested. The scores
for these seniors are as follow:
19 24 23 27 19 20 27 21 18
On the basis of this sample, can the psychologist
conclude that this year’s class has a different level
of optimism than last year’s class?
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Inference about Means and Mean Differences
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STEP-1: State the Hypothesis, and
select an alpha level
 H0 : μ = 19
 H1 : μ ≠ 19
(there is no change)
(this year’s mean is different)
 Example we use α = .05 two tail
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Inference about Means and Mean Differences
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STEP-2: Locate the critical region
 Remember that for hypothesis test with t
statistic, we must consult the t distribution
table to find the critical t value. With a sample
of n = 9 students, the t statistic will have
degrees of freedom equal to
df = n – 1 = 9 – 1 = 8
 For a two tailed test with α = .05 and df = 8, the
critical values are t = ± 2.306. The obtained t
value must be more extreme than either of
these critical values to reject H0
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Inference about Means and Mean Differences
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STEP-3: Obtain the sample data, and
compute the test statistic
SM =
t=
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s
√n
M-μ
SM
 Find the sample mean
 Find the sample
variances
 Find the estimated
standard error SM
 Find the t statistic
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Inference about Means and Mean Differences
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STEP-4: Make a decision about H0,
and state conclusion
 The obtained t statistic (t = 2.626) is in the
critical region. Thus our sample data are
unusual enough to reject the null
hypothesis at the .05 level of significance.
 We can conclude that there is a significant
difference in level of optimism between this
year’s and last year’s graduating classes
t(8) = 2.626, p<.05, two tailed
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Inference about Means and Mean Differences
The critical region in the
t distribution for α = .05 and df = 8
Reject H0
Reject H0
Fail to reject H0
-2.306
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2.306
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Inference about Means and Mean Differences
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DIRECTIONAL HYPOTHESES AND
ONE-TAILED TEST
 The non directional (two-tailed) test is more
commonly used than the directional (onetailed) alternative
 On other hand, a directional test may be
used in some research situations, such as
exploratory investigation or pilot studies or
when there is a priori justification (for
example, a theory previous findings)
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Inference about Means and Mean Differences
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LEARNING CHECK
A fund raiser for a charitable organization
has set a goal of averaging at least $ 25 per
donation. To see if the goal is being met, a
random sample of recent donation is
selected.
The data for this sample are as follows:
20 50 30 25 15 20 40 50 10 20
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Inference about Means and Mean Differences
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The critical region in the
t distribution for α = .05 and df = 9
Reject H0
Fail to reject H0
1.883
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Inference about Means and Mean Differences
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Chapter 10
THE t TEST FOR TWO
INDEPENDENT SAMPLES
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Inference about Means and Mean Differences
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Preview-2
 In many research situations, however, its
difficult or impossible for a researcher to
satisfy completely the rigorous requirement
of an experiment
 In these situations, a researcher can often
devise a research strategy (a method of
collecting data) that is similar to an
experiment but fails to satisfy at least one of
the requirement of a true experiment
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Inference about Means and Mean Differences
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NonExperimental and Quasi Experimental
 Although these studies resemble
experiment, they always contain a
confounding variable or other threat to
internal validity that is an integral part of
the design and simply cannot be removed
 The existence of a confounding variable
means that these studies cannot establish
unambiguous cause-and-effect relationship
and, therefore, are not true experiment
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Inference about Means and Mean Differences
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NonExperimental and Quasi Experimental
 … is the degree to which the research
strategy limits the confounding and control
threats to internal validity
 If a research design makes little or no
attempt to minimize threats, it is classified
as nonexperimental
 A quasi experimental design makes some
attempt to minimize threats to internal
validity and approach the rigor of a true
experiment
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Inference about Means and Mean Differences
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In an experiment…
 … a researcher typically creates treatment
condition by manipulating an IV, then
measures participants to obtain a set of
scores within each condition
 If the score in one condition are
significantly different from the other score
in another condition, the researcher can
conclude that the two treatment condition
have different effects
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Inference about Means and Mean Differences
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NonExperimental and Quasi Experimental
 Similarly, a nonexperimental study also
produces group of scores to be compared
for significant differences
 One variable is used to create groups or
conditions, then a second variable is
measured to obtain a set of scores within
each condition
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Inference about Means and Mean Differences
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NonExperimental and Quasi Experimental
 In nonexperimental and quasi-experimental
studies, the different groups or conditions are
not created by manipulating an IV
 The groups usually defined in terms of a
preexisting participant variable (male/female)
or in term of time (before/after)
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Inference about Means and Mean Differences
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 Single sample techniques are used occasionally in
real research, most research studies require the
comparison of two (or more) sets of data
 There are two general research strategies that can
be used to obtain of the two sets of data to be
compared:
○ The two sets of data come from the two completely
separate samples (independent-measures or
between-subjects design)
○ The two sets of data could both come from the
same sample (repeated-measures or within
subject design)
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Inference about Means and Mean Differences
Do the achievement
scores for students
taught by method A
differ from the scores
for students taught
by method B?
In statistical terms,
are the two
population means the
same or different?
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Taught by
Method A
Taught by
Method B
Unknown
µ =?
Unknown
µ =?
Sample
A
Sample
B
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Inference about Means and Mean Differences
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THE HYPOTHESES FOR AN
INDEPENDENT-MEASURES TEST
 The goal of an independent-measures
research study is to evaluate the mean
difference between two population (or
between two treatment conditions)
H0: µ1 - µ2 = 0 (No difference between the
population means)
H1: µ1 - µ2 ≠ 0 (There is a mean difference)
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Inference about Means and Mean Differences
THE FORMULA FOR AN INDEPENDENTMEASURES HYPOTHESIS TEST
t=
sample mean
difference
-
population mean
difference
estimated standard error
=
M1 – M2
S (M1 – M2)
 In this formula, the value of M1 – M2 is obtained
from the sample data and the value for µ1 - µ2
comes from the null hypothesis
 The null hypothesis sets the population mean
different equal to zero, so the independentmeasures t formula can be simplifier further
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Inference about Means and Mean Differences
THE STANDARD ERROR
To develop the formula for S(M1 – M2) we will
consider the following points:
 Each of the two sample means represent its
own population mean, but in each case
there is some error
SM =
√
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2
s
n
SM1-M2 =
√
s1
2
s2
2
+
n1
n2
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Inference about Means and Mean Differences
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POOLED VARIANCE
 The standard error is limited to situation in
which the two samples are exactly the same
size (that is n1 – n2)
 In situations in which the two sample size
are different, the formula is biased and,
therefore, inappropriate
 The bias come from the fact that the formula
treats the two sample variance
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Inference about Means and Mean Differences
POOLED VARIANCE
 for the independent-measure t statistic,
there are two SS values and two df values
SP
2 = SS
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n
SM1-M2 =
√
s1
2
s2
2
+
n1
n2
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Inference about Means and Mean Differences
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HYPOTHESIS TEST WITH THE
INDEPENDENT-MEASURES t STATISTIC
In a study of jury behavior, two samples of
participants 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 participants were asked to
recommend a jail sentence. The length of term
suggested by each participant is presented. Is
there a significant difference between the two
groups in their responses?
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Inference about Means and Mean Differences
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THE LENGTH OF TERM SUGGESTED
BY EACH PARTICIPANT
Group-1 scores:
Group-2 scores:
4 4 3 2 5 1 1 4
3 7 8 5 4 7 6 8
There are two separate samples in this
study. Therefore the analysis will use
the independent-measure t test
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Inference about Means and Mean Differences
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STEP-1: State the Hypothesis, and
select an alpha level
 H0 : μ1 - μ2 = 0 (for the population, knowing
evidence has been withheld has no effect on
the suggested sentence)
 H1 : μ1 - μ2 ≠ 0 (for the population,
knowledge of withheld evidence has an
effect on the jury’s response)
 We will set α = .05 two tail
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Inference about Means and Mean Differences
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STEP-2: Identify the critical region
 For the independent-measure t statistic,
degrees of freedom are determined by
df = n1 + n2 – 2 = 8 + 8 – 2 = 14
 The t distribution table is consulted, for a
two tailed test with α = .05 and df = 14, the
critical values are t = ± 2.145.
 The obtained t value must be more extreme
than either of these critical values to reject
H0
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Inference about Means and Mean Differences
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STEP-3: Compute the test statistic
 Find the sample mean for each group
M1 = 3 and M2 = 6
 Find the SS for each group
SS1 = 16 and SS2 = 24
 Find the pooled variance, and
SP2 = 2.86
 Find estimated standard error
S(M1-M2) = 0.85
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Inference about Means and Mean Differences
STEP-3: Compute the t statistic
t=
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M1 – M2
S (M1 – M2)
=
-3
0.85
= -3.55
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Inference about Means and Mean Differences
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STEP-4: Make a decision about H0,
and state conclusion
 The obtained t statistic (t = -3.53) is in the
critical region on the left tail (critical t = ±
2.145). Therefore, the null hypothesis is
rejected.
 The participants that were informed about
the withheld evidence gave significantly
longer sentences,
t(14) = -3.55, p<.05, two tails
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 
Inference about Means and Mean Differences
The critical region in the
t distribution for α = .05 and df = 14
Reject H0
Reject H0
Fail to reject H0
-2.145
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2.145
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Inference about Means and Mean Differences
LEARNING CHECK
The following data are from two separate
independent-measures experiments. Without doing
any calculation, which experiment is more likely to
demonstrate a significant difference between
treatment A and B? Explain your answer.
EXPERIMENT A
EXPERIMENT B
Treatment A Treatment B Treatment A Treatment B
n = 10
M = 42
SS = 180
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n = 10
M = 52
SS = 120
n = 10
M = 61
SS = 986
n = 10
M = 71
SS = 1042
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Inference about Means and Mean Differences
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LEARNING CHECK
A psychologist studying human memory,
would like to examine the process of
forgetting. One group of participants is
required to memorize a list of words in the
evening just before going to bed. Their
recall is tested 10 hours latter in the
morning. Participants in the second group
memorized the same list of words in he
morning, and then their memories tested
10 hours later after being awake all day.
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Inference about Means and Mean Differences
LEARNING CHECK
The psychologist hypothesizes that there will
be less forgetting during less forgetting during
sleep than a busy day. The recall scores for two
samples of college students are follows:
Asleep Scores
Awake Scores
15
13
14
14
15
13
14
12
16
15
16
15
14
13
11
12
16
15
17
14
13
13
12
14
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Inference about Means and Mean Differences
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LEARNING CHECK
 Sketch a frequency distribution for the ‘asleep’
group. On the same graph (in different color),
sketch the distribution for the ‘awake’ group.
Just by looking at these two distributions,
would you predict a significant differences
between two treatment conditions?
 Use the independent-measures t statistic to
determines whether there is a significant
difference between the treatments. Conduct
the test with α = .05
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Inference about Means and Mean Differences
 
Chapter 11
THE t TEST FOR TWO
RELATED SAMPLES
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Inference about Means and Mean Differences
 
OVERVIEW
 With a repeated-measures design, two sets
of data are obtained from the same sample
of individuals
 The main advantage of a repeated-measures
design is that it uses exactly the same
individual in all treatment conditions.
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Inference about Means and Mean Differences
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The Hypotheses for a Related-Samples Test
 As always, the null hypotheses states that
for the general population there is no effect,
no change, or no difference.
H0: X2 - X1 = μD = 0
 The alternative hypotheses states that there
is a treatment effect that causes the scores in
one treatment condition to be systematically
higher (or lower) than the scores in the
other condition. In symbols H1: μD ≠ 0
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Inference about Means and Mean Differences
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The t Statistic for Related Samples
 The t statistic for related samples is
structurally similar to the other t statistics
 One major distinction of the related samples
t is that is based on difference scores rather
than raw scores (X values)
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Inference about Means and Mean Differences
The t Statistic for Related Samples
t=
sample
population
statistic
parameter
estimated standard error
-
SMD =
S2
√ n-1
S2 =
© aSup-2007
or SMD =
SS
n-1
=
=
MD – μD
SMD
S
√ df
SS
df
or S =
√
SS
df
Inference about Means and Mean Differences
 
LEARNING CHECK
People with agoraphobia are so filled with anxiety
about being in public places that they seldom leave
their homes. Knowing this is a difficulty disorder to
treat, a researcher tries a long-term treatment.
A sample of individuals report how often they
have ventured out of the house in the past month.
Then they have receive relaxation training and are
introduce to trips away from the house at gradually
increasing durations.
After 2 months of treatment, participants report the
number of trip out of the house they made in the
last 30 days.
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Inference about Means and Mean Differences
Person
Before (X1)
After (X2)
 
Difference (D)
A
4
0
?
B
0
0
?
C
14
3
?
D
23
3
?
E
9
2
?
F
8
0
?
G
6
0
?
Does the treatment have a significant effect on
the number of trips a person takes?
Test with α = .05 two tails
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