Using Power to Determine Sample Size
whether the diet served at the college
cafeteria is causing excess weight gain.
The effect (lbs) is the mean weight gain
(lbs) of first-year student residents after
one semester of living in a dormitory.
To determine the number of students
that ought to be included in the sample,
Power
We are planning a study to determine
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
the following power-plot was generated.
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Effect
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1. What null and alternative hypotheses are being tested?
a. H0: effect = 0 vs. HA: effect > 0
b. H0: effect ≤ 0 vs. HA: effect > 0
c. Choices (a) and (b) are equivalent, and both are “correct”. Choice (a) emphasizes that 0 is
the hypothetical value of the parameter effect to which the observed value will be
compared in the test, i.e., in the test statistic and in calculation of the P-value.
2. What is the effect (lbs),
is true?
a. 0
b. 2
c. 4
d. 6
e. 8
Power
numerically, if the null hypothesis
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0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
Effect
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3. What is the probability
(numerically) of rejecting the
hypothesis is true.
a. 0.01
b. 0.05
c. 0.10
d. 0.90
e. 0.95
Power
null hypothesis if the null
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0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
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4
6
Effect
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4. What (numerically) is the Type 1
a. 0.01
b. 0.05
c. 0.10
d. 0.90
e. 0.95
Power
error rate shown in the plot?
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
Effect
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5. What significance level is shown
in the plot?
b. 0.05
c. 0.10
d. All of the above
e. None of the above
Power
a. 0.01
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0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
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4
6
Effect
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6. The sample sizes used in the graph
are 32, 64, 128, and 256 students.
their sample size. Which curve is
based on the least sample size of
32?
a. orange
b. blue
c. green
d. red
e. not enough information is
given
Power
Label each of the four curves with
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
Effect
8
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7. For the sample of 32 students (the
red curve), what (numerically) is
alternative that that the effect is 6
lbs?
a. 0.05
b. 0.20
c. 0.40
d. 0.60
e. 0.80
Power
the power of the test against the
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0.9
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0.5
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0.3
0.2
0.1
0
0
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4
6
Effect
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8. For the sample of 32 students (the
red curve), what (numerically) is
against the alternative that that the
effect is 8 lbs?
a. 0.05
b. 0.20
c. 0.40
d. 0.60
e. 0.80
Power
the Type 2 error rate of the test
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
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0.1
0
0
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6
Effect
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For detecting an effect of 6 lbs, if
the maximum acceptable Type 2
error rate is 0.15 (i.e., 15%), why
would a sample size of 32 (red
curve) be too small?
a. Type 2 error rate would be
0.40, too big, larger than 0.15
b. Type 2 error rate would be
0.60, too big, larger than 0.15
c. Type 2 error rate would be
Power
9.
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0.0.05, too small, less than
0.15
d. Type 2 error rate would be 0.10, too small, less than 0.15
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Effect
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10. For detecting an effect of 6 lbs, if
the maximum acceptable Type 2
would a sample size of 256
(orange) be too large?
a. A sample of size 64 (green)
would be sufficient, with
power = 0.85 and Type 2 error
rate = 0.15
b. A sample of size 256 would
Power
error rate is 0.15 (i.e., 15%), why
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0.9
0.8
0.7
0.6
0.5
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0.1
0
0
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4
be wasting resources, because
the power would be greater than 0.99, far more than required.
c. Both of the above
d. None of the above
6
Effect
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