STATISTICS 1403
PROBABILITY AND STATISTICS FOR BIOSCIENCES
PROBLEM SET: PAIRED TEST AND POWER
Abstract. Solve these problems, and prepare a clean, annotated copy of
them.
1. A Paired t Test
Researchers studied the efficacy of saliva continine as an indicator for exposure
to tobacco smoke. Seven subjects–none heavy smokers–abstained from smoking for
a week, and then smoked a single cigarette. Continine levels in their saliva were
measured 12 and 24 hours after the exposure:
continine levels, nmol/l
subject after 12 hours after 24 hours
1
72
24
2
58
27
3
67
49
4
93
59
5
33
0
6
18
11
7
147
43
Test the research hypothesis that the continine level decreases over time at the
α = 0.01 level.
Useful Formulas.
di = y1,i − y2,i
t= p
d¯
s2d /n
2. Power Smoking
It turns out this experiment was only a pilot study whose main purpose was to
estimate the mean difference and the variance of differences. Use the variance
information calculated from the previous problem to estimate the number of
subjects needed to conduct the test and detect a mean difference of µd = 30 with
Type I error α = 0.01 and power 1 − β = 0.95. Use z-value critical values (rather
than t values) to simplify your calculations.
Useful Formula.
σ 2 (z1−α − zβ )2
n=
(µa − µ0 )2
1