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.
4
6
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. The sample sizes used in the graph
Label each of the four curves with
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
are 32, 64, 128, and 256 students.
1
0.9
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0.7
0.6
0.5
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0.1
0
0
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Effect
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3. Which of the four sample sizes
would be optimal for detecting an
acceptable power is 0.90 (i.e.,
90%)?
a. 32 (red)
b. 64 (green)
c. 128 (blue)
d. 256 (orange)
Power
effect of 5 lbs, if the minimum
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0.9
0.8
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0.6
0.5
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0.3
0.2
0.1
0
0
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Effect
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For detecting an effect of 5 lbs, if
the minimum acceptable power is
0.90 (i.e., 90%), why would a
sample size of 32 be too small
a. Power would be 0.45, too
small, smaller than 0.90
b. Power would be 0.55, too
small, smaller than 0.15
c. Both of the above
d. None of the above
Power
4.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
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Effect
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12
For detecting an effect of 5 lbs, if
the minimum acceptable power is
0.90 (i.e., 90%), why would a
sample size of 256 be too large?
a. A sample of size 128 (blue)
would be sufficient, with
power = 0.93 and Type 2 error
rate = 0.17
b. A sample of size 256 would
be wasting resources, because
the power would be greater
than required.
c. Both of the above
d. None of the above
Power
5.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
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Effect
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6. Which of the four sample sizes
would be optimal for detecting an
acceptable Type 2 error rate is
0.15 (i.e., 15%)?
a. 32 (red)
b. 64 (green)
c. 128 (blue)
d. 256 (orange)
Power
effect of 6 lbs, if the maximum
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
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Effect
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Would increasing the significance
level from 0.05 to 0.10 increase,
decrease, or not change the Type 2
error rate?
a. increase
b. decrease
c. not change
Power
7.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
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Effect
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Would decreasing the Type 1
error rate from 0.05 to 0.01
increase, decrease, or not change
the Type 2 error rate?
a. increase
b. decrease
c. not change
Power
8.
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0.8
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0.5
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0
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Effect
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We could redesign the study by
observing only one sex, male or
female. While this would limit the
scope of inference to a narrower
population, it would decrease the
standard deviation of the
underlying random variable of
interest, weight gain (lbs) = effect
(lbs). Would decreasing the
standard deviation increase,
decrease, or not change the Type 2
error rate?
a. increase
b. decrease
c. not change
Power
9.
1
0.9
0.8
0.7
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0.5
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0.2
0.1
0
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Effect
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10. Would changing the Type 2 error
rate (by changing the sample size,
and/or the standard deviation)
change the Type 1 error rate?
a. Yes
b. No, the Type 1 error rate is
fixed at the set value of the
significance level.
Power
the minimum effect to detect,
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
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Effect
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