Transcript Document

Chapter 11
Statistical Inference: OneSample Confidence Interval
I Criticisms of Null Hypothesis Significance
Testing
 Does not indicate whether the effect is large or
small
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 Answers the wrong question: Prob(D|H0). The
correct question concerns Prob(H0|D).
 Is a trivial exercise; all null hypotheses are false.
 Turns a continuum of uncertainty into a reject-donot reject decision.
II Confidence Interval for 
 A confidence interval for  is a segment on the
real number line such that that  has a high
probability of lying on the segment.
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f (t ) /2 = .05/2
/2 = .05/2
1   = .95
t
 t.05/2, 
0
t.05/2, 
Figure 1. Sampling distribution of t. If one t statistic is randomly
sampled from this population of t’s, the probability is .95 that the
obtained t will come from the interval from –t.05/2,  to t.05/2, .
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1. From Figure 1, the following probability statement
follows:
Prob(t.05/ 2,   t  t.05/ 2,  )  1  
2. Replacing t with ( X   ) / (φ / n ) and using
some algebra gives the following 100(1 – )%
two-sided confidence interval for 
t  / 2, φ
t  / 2, φ
Pr ob(X 
 X 
)  1 
1 44 2 4 n43
1 44 2 4n 43
L1
L2
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3. L1 and L2 denote, respectively, the lower and
upper endpoints of the open confidence interval
for .
4. A researcher can be 100(1 – )% confident that 
is greater than L1 and less than L2.
5. The probability (1 – ) is called the confidence
coefficient and is usually equal to (1 – .05 ) = .95.
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6. The assumptions associated with a confidence
interval are the same as those for a one-sample t
statistic.
A. Computational Example: Two-Sided Interval
1. Consider the following hypotheses for the
Idle-On-In College registration example:
H0:  = 0
H1:  ≠ 0
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2. A two-sided 100(1 – .05) = 95% confidence
interval for , where X = 2.90, φ = 0.3013,
and t.05/ 2,26  2.056, is
X
t .05/ 2,26φ
n
2.90 
2.056(0.3013)
t .05/ 2,26φ
 X 
)
n
   2.90 
27
2.056(0.3013)
27
2.78    3.02
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3. The dean can be 100(1 – .05) = 95% confident
that  is greater than 2.78 and less than 3.02.
4. The dean can be even more confident that  lies
in the interval from L1 to L2 by computing a
100(1 – .01) = 99% confidence interval.
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5. A two-sided 100(1 – .01) = 99% confidence
interval for , where t.01/2, 26 = 2.779, is given by
X
t .01/ 2,26φ
n
2.90 
2.779(0.3013)
t .01/2,26φ
 X 
)
n
   2.90 
27
2.779(0.3013)
27
2.74    3.06
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6. Graphs of the two confidence intervals
L1 = 2.78
2.6
2.7
2.8
L2 = 3.02
2.9
3.0
3.1
3.2
95% confidence interval for 
L1 = 2.74
2.6
2.7
2.8
L2 = 3.06
2.9
3.0
3.1
3.2
99% confidence interval for 
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7. As the dean’s confidence that she has captured 
increases, so does the size of the interval from L1
to L2.
B. More On the Interpretation of Confidence
Intervals
C. Computational Example: One-Sided Interval
1. Suppose that one-tailed hypotheses, H0:  ≥ 0
and H1:  < 0, reflect the dean’s hunch about
the new registration procedure.
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2. A one-sided 100(1 – .05) = 95% confidence
interval for , where X = 2.90, φ = 0.3013,
and t.05,26  1.706, is
t .05,26φ
 X 
)
n
  2.90 
1.706(0.3013)
27
  3.00
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3. Comparison of one- and two-sided confidence
intervals
L2 = 3.00
2.6
2.7
2.8
2.9
3.0
3.1
3.2
One-sided 95% confidence interval for 
L1 = 2.78
2.6
2.7
2.8
L2 = 3.02
2.9
3.0
3.1
3.2
Two-sided 95% confidence interval for 
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D. Advantages of Confidence Interval Estimation
Over Hypothesis Testing
1. Hypothesis testing is not very informative. A
confidence interval narrows the range of possible
values for .
2. Confidence intervals can be used to test all null
hypotheses such as H0:  = 0. Any 0 that lies
outside of the confidence interval corresponds to a
rejectable null hypothesis.
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3. A sample mean and confidence interval provide
an estimate of the population parameter and a
range of values—the error variation—qualifying
the estimate.
4. A 100(1 – )% confident interval for  contains
all of the values of 0 for which the null
hypothesis would not be rejected.
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III Practical Significance
A. Estimator of Cohen’s d
1. Hedges’s g for the registration example
| X  0 | |12.90  3.10 |
g

 0.66
φ
0.3013
2. Interpretation of g
 g = 0.2 is a small effect
 g = 0.5 is a medium effect
 g = 0.8 is a large effect
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3. Computation of g from t statistics in research
reports
g
t
n
4. For the registration example, t = 3.449 and n = 27
g
t
n

3.449
 0.66
27
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