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2. Sampling distribution of t
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The distribution of t values that would be
obtained if a value of t were calculated for
each sample mean for all possible random
samples of a given size from a population
t ratio:
Sampling distribution of t
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_
t=
The result is a family of distributions, each
associated with a specific degree of freedom
Properties:
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(X - µhyp)
s
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__
x
Sampling distribution of t
Sampling distribution of t
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Degrees of freedom
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Does the mean gas mileage for some
population of cars drop below the legally
required minimum of 45 miles per gallon?
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Sample:
n = 6 cars
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X = 43; s = 2.19
the number of values free to vary, given one or
more mathematical restrictions on a sample of
observed values used to estimate some unknown
population characteristic
Degrees of freedom (single sample)
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3. Example: Gas mileage
investigation
Symmetrical, unimodal, bell-shaped (similar to the normal
curve)
For small values of df, tails are inflated, and the mean is
smaller than the mean of a corresponding normal curve
It gets closer and closer to a normal curve as the number
of df increases
df = n - 1
II. Inferential Statistics (8)
!  Basics
of Experimentation
for a difference between means
!  t test for two independent samples
!  Testing
1
2. Testing for a difference
between means
1.- Basics of Experimentation
!  Experiment
• 
independent & dependent variables
!  Looking for cause-effect explanations
!  The simplest case: Comparing two groups
(samples)
Two independent samples
!  Control,
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Two related samples
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Statistical Hypotheses
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Nondirectional
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Directional (lower tail critical):
H0: µ1 - µ2 ≥ 0
H1: µ1 - µ2 < 0
Directional (upper tail critical):
H 0: µ 1 - µ 2 ≤ 0
H1: µ1 - µ2 > 0
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Testing for a difference between
means
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Standard error of X1 – X2
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A rough measure of the average amount by which
any difference between sample means deviates
from the difference between population means
Any difference between two population means
Testing for a difference between
means
H0: µ1 - µ2 = 0
H1: µ1 - µ2 ≠ 0
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Observations in one sample are paired, on a one-toone basis, with observations in the other sample
Effect
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Testing for a difference between
means
Observations in one sample are not paired, on a oneto-one basis, with observations in the other sample
Sampling distribution of X1 – X2
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Differences between sample means based on all
possible pairs of random samples (of given sizes)
from two underlying populations
3. t test for two independent
samples
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t ratio for two populations means (two
independent samples)
… as always:
Statistic - Parameter
Test statistic =
Standard error of the statistic
t=
(X1 – X2) – (µ1 – µ2)
s X1 – X2
2
t test for two independent
samples
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Estimated standard error of the difference
between sample means
t test for two independent
samples
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Example: see pdf file
sX1 – X2 = √[s2 (1/n1 + 1/n2)]
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Pooled estimate of variance
(SS1 + SS2)
s2p =
n1 + n2 - 2
4. Other related concepts
!  The
!  p-values
!  Statistical
significance
Effect Size
!  Estimating
!  Cohen’s
d
Statistical significance
!  Tests
p-values
of hypotheses are often referred to as tests
of significance. Test results are described as
!  statistically significant if the H0 has been
rejected
!  not statistically significant if the H0 has been
retained
degree of rarity of a test result, given that the
null hypothesis is true
!  Smaller p-values tend to discredit the null
hypothesis and to support the research hypothesis
Statistical significance
!  Implies
only that H0 is
probably false, and not
whether it is false
because of a large or
small difference
between population
means (effect size)
3
Estimating Effect Size
Estimating Effect Size: Cohen’s d
!  Confidence
intervals for µ1-µ2
of values that, in the long run, include
the unknown effect a certain percent of the
time
!  Ranges
!  CI
for µ1-µ2 (two independent samples)
!  X1-X2 ± (tconf)(sX1-X2)
!  Spanish learning example (see PDF notes):
-19.44 ± 2.13 * 5.5 ; [-31.16, -7.73]
√s2p
19.44
19.44
=
√128.07
11.32
=
=
1.72
Estimating Effect Size: Cohen’s d
!  Cohen’s
guidelines for d
Effect size is:
!  small if d < 0.2 or d ≈ 0.2
!  medium if d ≈ 0.5
!  large if d > 0.8 or d ≈ 0.8
II. Inferential Statistics (9)
1.- Analysis of Variance
!  Analysis
!  ANOVA
!  One
of Variance
way ANOVA
!  Between-groups design
!  Effect size
!  Multiple comparisons
X1-X2
st. deviation
√s2p
st. deviation supplies a stable frame of
reference not influenced by increases in
sample size
d , the Standardized Effect Estimate, for
the Spanish learning example (see PDF notes):
=
mean difference
!  The
!  Cohen’s
|X1-X2|
d , the Standardized Effect Estimate
effect size by expressing the
observed mean difference in standard
deviation units
!  Describes
d=
Estimating Effect Size: Cohen’s d
d=
!  Cohen’s
!  An
overall test of the null hypothesis for more
than two population means
!  Null
Hypotheses
!  H0
: µ0 = µ1 = µ2
!  One-way
ANOVA
!  The
simplest type, that tests whether differences
exist among population means categorized by
only one factor or independent variable
4
Analysis of Variance (ANOVA)
!  Two
sources of variability
!  Variability
between groups
•  Variability among scores of subjects who, being in
different groups, receive different experimental
treatments
!  Variability
within groups
•  Variability among scores of subjects who, being in the
same group, receive the same experimental treatments
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