Biostatistics 22S:101 Answers to Practice Exam 2
1. A large population contains an unknown proportion (p) of black marbles. A sample of n=200 drawn scientifically
from the population contained x=40 black marbles. Which picture shows the posterior distribution of the population
proportion p? Justify your answer. A. Because sep = sqrt(.2*.8/200) = .028
A. Looks like this (µ = .2 and σ = .028) B. Looks like this (µ= .2 and σ = .089)
0.30 0.12
0.24
0.18
0.06
0.12
0.06
0.00 0.00
0 0.5 1 0 0.5 1
2. An investigator wants to determine the proportion (p) of retirees who chose not to fill a prescription last year because it
was too expensive. In a scientific sample of n=525, x = 98 respondents said that they had done this. State in words the
posterior distribution of p. Obtain a 95% credible interval for p. What (approximately) is the probability that p exceeds
.25 ? 95% CI: .153 to .220. P(p > .25 | data ) = (Area above z = 3.72) = .0000.
3. A prospective study recruited 10,000 smokers and 10,000 non-smokers aged 30 to 39 and followed them for 20 years.
The cases of throat cancer are shown in this table.
no Cancer Cancer Relative
Frequency(%)
Smokers 9800 200 2.0%
Non-Smokers 9950 50 0.5%
What is the estimated relative risk of throat cancer for smokers vs. non-smokers? m = 2.0/.5 = 4.0
RR
4. Two hundred forty dieters volunteered for a study of chromium picolinate, a fat-reducing dietary supplement. They
were randomly assigned to receive placebo or chromium picolinate. One side effect is reduction in iron, a key component
of hemoglobin. Here are the data on percents of subjects with lower iron after 8 weeks of treatment.
Chromium
Placebo Picolinate
n 120 120
% with reduced iron 16% 29%
sep 0.033 0.041
∆ˆ 0.130
sed 0.053
95% CI 0.026 0.234
• What are the mean difference and the standard error of the difference (SED) between the percents?
• Obtain a 95% credible interval for the difference.
• Is the difference statistically significant? Yes.
5. In a study to compare osteoporosis rates for men and women over the age of 70 it was observed that 6.9% of 25000
men and 67% of 27000 women had osteoporosis. The difference is 60.1 percentage points and the credible interval is
59.4 to 60.7. Is the difference significant? Yes, zero is ruled out.
1. A large population contains an unknown proportion (p) of black marbles. A sample of n=200 drawn scientifically
from the population contained x=40 black marbles. Which picture shows the posterior distribution of the population
proportion p? Justify your answer. A. Because sep = sqrt(.2*.8/200) = .028
A. Looks like this (µ = .2 and σ = .028) B. Looks like this (µ= .2 and σ = .089)
0.30 0.12
0.24
0.18
0.06
0.12
0.06
0.00 0.00
0 0.5 1 0 0.5 1
2. An investigator wants to determine the proportion (p) of retirees who chose not to fill a prescription last year because it
was too expensive. In a scientific sample of n=525, x = 98 respondents said that they had done this. State in words the
posterior distribution of p. Obtain a 95% credible interval for p. What (approximately) is the probability that p exceeds
.25 ? 95% CI: .153 to .220. P(p > .25 | data ) = (Area above z = 3.72) = .0000.
3. A prospective study recruited 10,000 smokers and 10,000 non-smokers aged 30 to 39 and followed them for 20 years.
The cases of throat cancer are shown in this table.
no Cancer Cancer Relative
Frequency(%)
Smokers 9800 200 2.0%
Non-Smokers 9950 50 0.5%
What is the estimated relative risk of throat cancer for smokers vs. non-smokers? m = 2.0/.5 = 4.0
RR
4. Two hundred forty dieters volunteered for a study of chromium picolinate, a fat-reducing dietary supplement. They
were randomly assigned to receive placebo or chromium picolinate. One side effect is reduction in iron, a key component
of hemoglobin. Here are the data on percents of subjects with lower iron after 8 weeks of treatment.
Chromium
Placebo Picolinate
n 120 120
% with reduced iron 16% 29%
sep 0.033 0.041
∆ˆ 0.130
sed 0.053
95% CI 0.026 0.234
• What are the mean difference and the standard error of the difference (SED) between the percents?
• Obtain a 95% credible interval for the difference.
• Is the difference statistically significant? Yes.
5. In a study to compare osteoporosis rates for men and women over the age of 70 it was observed that 6.9% of 25000
men and 67% of 27000 women had osteoporosis. The difference is 60.1 percentage points and the credible interval is
59.4 to 60.7. Is the difference significant? Yes, zero is ruled out.
