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Inference about Means and Mean Different
 
PART III
Inference about
Means and Mean
Different
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Inference about Means and Mean Different
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Chapter 8
INTRODUCTION TO
HYPOTHESIS TESTING
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Inference about Means and Mean Different
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The Logic of Hypothesis Testing
 It usually is impossible or impractical for a
researcher to observe every individual in a
population
 Therefore, researchers usually collect data
from a sample and then use the sample data
to answer question about the population
 Hypothesis testing is statistical method that
uses sample data to evaluate a hypothesis
about the population
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Inference about Means and Mean Different
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The Hypothesis Testing Procedure
1. State a hypothesis about population, usually the
hypothesis concerns the value of a population
parameter
2. Before we select a sample, we use hypothesis to
predict the characteristics that the sample have.
The sample should be similar to the population
3. We obtain a sample from the population
(sampling)
4. We compare the obtain sample data with the
prediction that was made from the hypothesis
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Inference about Means and Mean Different
PROCESS OF HYPOTHESIS TESTING
 It assumed that the parameter μ is known for the
population before treatment
 The purpose of the experiment is to determine
whether or not the treatment has an effect on the
population mean
Known population
before treatment
Unknown population
after treatment
TREATMENT
μ = 30
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μ=?
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Inference about Means and Mean Different
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EXAMPLE
 It is known from national health statistics
that the mean weight for 2-year-old children
is μ = 26 pounds and σ = 4 pounds
 The researcher’s plan is to obtain a sample
of n = 16 newborn infants and give their
parents detailed instruction for giving their
children increased handling and
stimulation
 NOTICE that the population after treatment
is unknown
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Inference about Means and Mean Different
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STEP-1: State the Hypothesis
 H0 : μ = 26 (even with extra handling, the
mean at 2 years is still 26 pounds)
 H1 : μ ≠ 26 (with extra handling, the mean
at 2 years will be different from 26 pounds)
 Example we use α = .05 two tail
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Inference about Means and Mean Different
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STEP-2: Set the Criteria for a Decision
 Sample means that are likely to be obtained
if H0 is true; that is, sample means that are
close to the null hypothesis
 Sample means that are very unlikely to be
obtained if H0 is false; that is, sample means
that are very different from the null
hypothesis
 The alpha level or the significant level is a
probability value that is used to define the
very unlikely sample outcomes if the null
hypothesis is true
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Inference about Means and Mean Different
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The location of the critical region
boundaries for three different los
-1.96
-2.58
-3.30
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α = .05
α = .01
α = .001
1.96
2.58
3.30
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Inference about Means and Mean Different
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STEP-3: Collect Data and Compute
Sample Statistics
 After obtain the sample data, summarize
the appropriate statistic
σM =
σ
√n
M-μ
z= σ
M
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NOTICE
 that the top f the z-scores formula
measures how much difference
there is between the data and the
hypothesis
 The bottom of the formula
measures standard distances that
ought to exist between the sample
mean and the population mean
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Inference about Means and Mean Different
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STEP-4: Make a Decision
 Whenever the sample data fall in the critical
region then reject the null hypothesis
 It’s indicate there is a big discrepancy
between the sample and the null hypothesis
(the sample is in the extreme tail of the
distribution)
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Inference about Means and Mean Different
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LEARNING CHECK
HYPOTHESIS TEST WITH z
 A standardized test that are normally
distributed with μ = 65 and σ = 15. The
researcher suspect that special training in
reading skills will produce a change in
scores for individuals in the population. A
sample of n = 25 individual is selected, the
average for this sample is M = 70.
 Is there evidence that the training has an
effect on test score?
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Inference about Means and Mean Different
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FACTORS THAT INFLUENCE A
HYPOTHESIS TEST
M-μ
z= σ
M
σM =
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σ
√n
 The size of difference
between the sample mean
and the original population
mean
 The variability of the
scores, which is measured
by either the standard
deviation or the variance
 The number of score in the
sample
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Inference about Means and Mean Different
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DIRECTIONAL (ONE-TAILED)
HYPOTHESIS TESTS
 Usually a researcher begin an experiment
with a specific prediction about the
direction of the treatment effect
 For example, a special training program is
expected to increase student performance
 In this situation, it possible to state the
statistical hypothesis in a manner that
incorporates the directional prediction into
the statement of H0 and H1
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Inference about Means and Mean Different
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LEARNING CHECK
A psychologist has developed a standardized
test for measuring the vocabulary skills of 4year-old children. The score on the test form a
normal distribution with μ = 60 and σ = 10.
A researcher would like to use this test to
investigate the hypothesis that children who
grow up as an only child develop vocabulary
skills at a different rate than children in large
family. A sample of n = 25 only children is
obtained, and the mean test score for this sample
is M = 63.
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Inference about Means and Mean Different
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Chapter 9
INTRODUCTION TO
t STATISTIC
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Inference about Means and Mean Different
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THE t STATISTIC:
AN ALTERNATIVE TO z
 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
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Inference about Means and Mean Different
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The statistical procedure were based
on a few basic concepts:
1. A sample mean (M) is expected more or less
to approximate its population mean (μ). This
permits us to use sample mean to test a
hypothesis about the population mean.
2. The standard error provide a measure of how
well a sample mean approximates the
population mean. Specially, the standard error
determines how much difference between M
and μ is reasonable to expect just by chance.
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Inference about Means and Mean Different
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The statistical procedure were based
on a few basic concepts:
3. To quantify our inferences about the
population, we compare the obtained
sample mean (M) with the hypothesized
population mean (μ) by computing a zscore test statistic
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Inference about Means and Mean Different
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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 Different
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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 Different
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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 Different
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 Different
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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 Different
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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 Different
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 Different
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
t=
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sample mean
(from data)
-
population mean
(hypothesized from H0)
Estimated standard error
(computed from the sample data)
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Inference about Means and Mean Different
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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 Different
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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 Different
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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 Different
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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 Different
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STEP-4: Make a decision about H0,
and state conclusion
 The obtained t statistic (t = -4.39) 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) = -4.39, p<.05, two tailed
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Inference about Means and Mean Different
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 Different
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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 Different
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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 Different
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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 Different
 
Chapter 10
THE t TEST FOR TWO
INDEPENDENT SAMPLES
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Inference about Means and Mean Different
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OVERVIEW
 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 Different
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 Different
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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 Different
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 independent-measures t formula
can be simplifier further
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Inference about Means and Mean Different
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 Different
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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 Different
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 Different
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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 Different
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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 Different
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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 Different
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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 Different
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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 Different
STEP-3: Compute the t statistic
t=
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M1 – M2
S (M1 – M2)
=
-3
0.85
= -3.53
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Inference about Means and Mean Different
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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) = -3.53, p<.05, two tails
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Inference about Means and Mean Different
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 Different
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 Different
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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 Different
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 Different
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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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