Practice Exercise Quiz 1
Question 1)
Consider influenza epidemics for two parent heterosexual
families.
families.
Suppose that the probability is 17% that at least one of the parents has
contracted the disease.
contracted the disease.
The probability that the father has contracted influenza is 12% while the
probability that both the mother and father have contracted the disease is
6%.
probability that both the mother and father have contracted the disease is
6%.
What is the probability that the mother has contracted influenza?
(Hints look at lecture 2 around 5:30 and homework question on page
3/10).
3/10).
 6%
 17%
 11%
 5%
Question 2)
A random variable, X is uniform, a box from 0 to 1 of height
1.
1.
(So that its density is f(x)=1 for 0≤x≤1.) What is its 75th
percentile?
percentile?
(Hints, look at lecture 2 around 21:30 and homework 1 page 4/10. Also, look
up the help function for the qunif command in R.)
up the help function for the qunif command in R.)
 0.10
 0.25
 0.75
 0.50
Question 3)
You are playing a game with a friend where you flip a coin and if it
comes up heads you give her X dollars and if it comes up tails she gives
you Y dollars. The probability that the coin is heads is p (some number
between 0 and 1.) What has to be true about X and Y to make so that both
of your expected total earnings is 0. The game would then be called
“fair”.
comes up heads you give her X dollars and if it comes up tails she gives
you Y dollars. The probability that the coin is heads is p (some number
between 0 and 1.) What has to be true about X and Y to make so that both
of your expected total earnings is 0. The game would then be called
“fair”.
(Hints, look at Lecture 4 from 0 to 6:50 and Homework 1 page 5/10. Also,
for further reading on fair games and gambling, start with the Dutch Book
problem ).
for further reading on fair games and gambling, start with the Dutch Book
problem ).
 X=Y
 p=X/Y
 p/1−p=Y/X
 p/1−p=X/Y
Question 4)
A density that looks like a normal density (but may or may not be exactly
normal) is exactly symmetric about zero. (Symmetric means if you flip it
around zero it looks the same.) What is its median?
normal) is exactly symmetric about zero. (Symmetric means if you flip it
around zero it looks the same.) What is its median?
(Hints, look at quantiles from Lecture 2 around 21:30 and the problem on
page 9/10 from Homework 1.)
page 9/10 from Homework 1.)
 The median must be 0.
 The median must be different from the mean.
 The median must be 1.
 We can’t conclude anything about the median.
Question 5)
Consider the following PMF shown below in R.
x < 1:4
p < x/sum(x)
temp < rbind(x, p)
rownames(temp) < c(“X”, “Prob”)
temp
## [,1] [,2] [,3] [,4]
## X 1.0 2.0 3.0 4.0
## Prob 0.1 0.2 0.3 0.4
What is the mean?
(Hint, watch Lecture 4 on expectations of PMFs and look at Homework 1
problem on page 10/10 for a similar problem calculating the variance.)
problem on page 10/10 for a similar problem calculating the variance.)
 4
 2
 1
 3
Question 6)
A web site (www.medicine.ox.ac.uk/bandolier/band64/b647.html) for home
pregnancy tests cites the following:
pregnancy tests cites the following:
“When the subjects using the test were women who collected and tested
their own samples, the overall sensitivity was 75%. Specificity was also
low, in the range 52% to 75%.” Assume the lower value for the
specificity.
their own samples, the overall sensitivity was 75%. Specificity was also
low, in the range 52% to 75%.” Assume the lower value for the
specificity.
Suppose a subject has a positive test and that 30% of women taking
pregnancy tests are actually pregnant.
pregnancy tests are actually pregnant.
What number is closest to the probability of pregnancy given the positive
test?
test?
(Hints, watch Lecture 3 at around 7 minutes for a similar
example. Also, there’s a lot of Bayes’ rule problems and descriptions
out there, for example here’s one for HIV testing. Note, discussions of
Bayes’ rule can get pretty heady. So if it’s new to you, stick to basic
treatments of the problem. Also see Homework 2 question on page
5/12.)
example. Also, there’s a lot of Bayes’ rule problems and descriptions
out there, for example here’s one for HIV testing. Note, discussions of
Bayes’ rule can get pretty heady. So if it’s new to you, stick to basic
treatments of the problem. Also see Homework 2 question on page
5/12.)
 20%
 30%
 40%
 10%
Statistical Inference Week 2 Quiz Answer
Practice Exercise Quiz 2
Question 1)
What is the variance of the distribution of the average an IID draw
of n observations from a population with mean μ and variance σ2.
of n observations from a population with mean μ and variance σ2.
Your Answer
Score
Explanation
Score
Explanation
 σ2
 σ/n
 2σ/n‾‾√
 σ2/n
Question 2)
Suppose that diastolic blood pressures (DBPs) for men aged 3544 are
normally distributed with a mean of 80 (mm Hg) and a standard
deviation of 10. About what is the probability that a random 3544
year old has a DBP less than 70?
normally distributed with a mean of 80 (mm Hg) and a standard
deviation of 10. About what is the probability that a random 3544
year old has a DBP less than 70?
 16%
 8%
 22%
 32%
Question 3)
Brain volume for adult women is normally distributed with a mean of
about 1,100 cc for women with a standard deviation of 75 cc. What brain
volume represents the 95th percentile?
about 1,100 cc for women with a standard deviation of 75 cc. What brain
volume represents the 95th percentile?
 approximately 1175
 approximately 977
 approximately 1223
 approximately 1247
Question 4)
Refer to the previous question. Brain volume for adult women is about
1,100 cc for women with a standard deviation of 75 cc. Consider the
sample mean of 100 random adult women from this population. What is
the 95th percentile of the distribution of that sample mean?
1,100 cc for women with a standard deviation of 75 cc. Consider the
sample mean of 100 random adult women from this population. What is
the 95th percentile of the distribution of that sample mean?
 approximately 1112 cc
 approximately 1115 cc
 approximately 1088 cc
 approximately 1110 cc
Question 5)
#You flip a fair coin 5 times, about what’s the probability of
getting 4 or 5 heads?
getting 4 or 5 heads?
 19%
 3%
 6%
 12%
Question 6)
The respiratory disturbance index (RDI), a measure of sleep
disturbance, for a specific population has a mean of 15 (sleep events
per hour) and a standard deviation of 10. They are not normally
distributed. Give your best estimate of the probability that a sample
mean RDI of 100 people is between 14 and 16 events per hour?
disturbance, for a specific population has a mean of 15 (sleep events
per hour) and a standard deviation of 10. They are not normally
distributed. Give your best estimate of the probability that a sample
mean RDI of 100 people is between 14 and 16 events per hour?
 68%
 95%
 34%
 47.5%
Question 7)
Consider a standard uniform density. The mean for this density is .5
and the variance is 1 / 12. You sample 1,000 observations from this
distribution and take the sample mean, what value would you expect it
to be near?
and the variance is 1 / 12. You sample 1,000 observations from this
distribution and take the sample mean, what value would you expect it
to be near?
 0.75
 0.10
 0.25
 0.5
Question 8)
#The number of people showing up at a bus stop is assumed to be
Poisson with a mean of 5 people per hour. You watch the bus stop for 3
hours. About what’s the probability of viewing 10 or fewer people?
Poisson with a mean of 5 people per hour. You watch the bus stop for 3
hours. About what’s the probability of viewing 10 or fewer people?
 0.03
 0.08
 0.12
 0.06
Statistical Inference Week 3 Quiz Answer
Practice Exercise Quiz 3
Question 1)
In a population of interest, a sample of 9 men yielded a sample
average brain volume of 1,100cc and a standard deviation of 30cc. What
is a 95% Students T confidence interval for the mean brain volume in
this new population?
average brain volume of 1,100cc and a standard deviation of 30cc. What
is a 95% Students T confidence interval for the mean brain volume in
this new population?
 [1092, 1108]
 [1080, 1120]
 [1031, 1169]
 [1077, 1123]
Question 2)
A diet pill is given to 9 subjects over six weeks. The average
difference in weight (follow up – baseline) is 2 pounds.
difference in weight (follow up – baseline) is 2 pounds.
What would the standard deviation of the difference in weight have to
be for the upper endpoint of the 95% T confidence interval to touch
0?
be for the upper endpoint of the 95% T confidence interval to touch
0?
 2.60
 1.50
 2.10
 0.30
Question 3
In an effort to improve running performance, 5 runners were either
given a protein supplement or placebo.
given a protein supplement or placebo.
Then, after a suitable washout period, they were given the opposite
treatment.
treatment.
Their mile times were recorded under both the treatment and placebo,
yielding 10 measurements with 2 per subject.
yielding 10 measurements with 2 per subject.
The researchers intend to use a T test and interval to investigate
the treatment.
the treatment.
Should they use a paired or independent group T test and interval?
 #It’s necessary to use both
 #A paired interval
 #You could use either
 #Independent groups, since all subjects were seen under both systems
Question 4)
In a study of emergency room waiting times, investigators consider a
new and the standard triage systems.
new and the standard triage systems.
To test the systems, administrators selected 20 nights and randomly
assigned the new triage system to be used on 10 nights and the
standard system on the remaining 10 nights. They calculated the
nightly median waiting time (MWT) to see a physician.
assigned the new triage system to be used on 10 nights and the
standard system on the remaining 10 nights. They calculated the
nightly median waiting time (MWT) to see a physician.
The average MWT for the new system was 3 hours with a variance of
0.60 while the average MWT for the old system was 5 hours with a
variance of 0.68. Consider the 95% confidence interval estimate for
the differences of the mean MWT associated with the new
system.
0.60 while the average MWT for the old system was 5 hours with a
variance of 0.68. Consider the 95% confidence interval estimate for
the differences of the mean MWT associated with the new
system.
Assume a constant variance. What is the interval? Subtract in this
order (New System – Old System).
order (New System – Old System).
 [1.29, 2.70]
 [1.25, 2.75]
 [2,70, 1.29]
 [2.75, 1.25]
Question 5)
Suppose that you create a 95% T confidence interval. You then create
a 90% interval using the same data. What can be said about the 90%
interval with respect to the 95% interval?
a 90% interval using the same data. What can be said about the 90%
interval with respect to the 95% interval?
 It is impossible to tell.
 The interval will be narrower.
 The interval will be wider
 The interval will be the same width, but shifted.
Question 6)
To further test the hospital triage system, administrators selected
200 nights and randomly assigned a new triage system to be used on 100
nights and a standard system on the remaining 100 nights. They
calculated the nightly median waiting time (MWT) to see a physician.
The average MWT for the new system was 4 hours with a standard
deviation of 0.5 hours while the average MWT for the old syste was 6 hours with
a standard deviation of 2 hours.
200 nights and randomly assigned a new triage system to be used on 100
nights and a standard system on the remaining 100 nights. They
calculated the nightly median waiting time (MWT) to see a physician.
The average MWT for the new system was 4 hours with a standard
deviation of 0.5 hours while the average MWT for the old syste was 6 hours with
a standard deviation of 2 hours.
Consider the hypothesis of a decrease in the mean MWT associated with
the new treatment.
the new treatment.
What does the 95% independent group confidence interval with unequal
variances suggest vis a vis this hypothesis?
variances suggest vis a vis this hypothesis?
#(Because there’s so many observations per group, just use the Z
quantile instead of the T.)
quantile instead of the T.)
 When subtracting (old – new) the interval is entirely above zero.
The new system appears to be effective.
 When subtracting (old – new) the interval contains 0. The new system
appears to be effective.
 When subtracting (old – new) the interval is entirely above zero.
The new system does not appear to be effective.
 When subtracting (old – new) the interval contains 0. There is not
evidence suggesting that the new system is effective.
Question 7)
Suppose that 18 obese subjects were randomized, 9 each, to a new diet
pill and a placebo.
pill and a placebo.
Subjects’ body mass indices (BMIs) were measured at a baseline and
again after having received the treatment or placebo for four
weeks.
again after having received the treatment or placebo for four
weeks.
The average difference from followup to the baseline (followup –
baseline) was −3 kg/m2 for the treated group and 1 kg/m2 for the
placebo group. The corresponding standard deviations of the
differences was 1.5 kg/m2 for the treatment group and 1.8 kg/m2 for
the placebo group. Does the change in BMI over the four week period
appear to differ between the treated and placebo groups?
baseline) was −3 kg/m2 for the treated group and 1 kg/m2 for the
placebo group. The corresponding standard deviations of the
differences was 1.5 kg/m2 for the treatment group and 1.8 kg/m2 for
the placebo group. Does the change in BMI over the four week period
appear to differ between the treated and placebo groups?
Assuming normality of the underlying data and a common population
variance, calculate the relevant *90%* t confidence interval.
variance, calculate the relevant *90%* t confidence interval.
Subtract in the order of (Treated – Placebo) with the smaller (more
negative) number first.
negative) number first.

 [5.364, 2.636]
 [5.531, 2.469]
 [2.469, 5.531]
 [2.636, 5.364]
Statistical Inference Week 4 Quiz Answer
Practice Exercise Quiz 4
Question 1)
A pharmaceutical company is interested in testing a potential blood
pressure lowering medication.
pressure lowering medication.
Their first examination considers only subjects that received the
medication at baseline then two weeks later.
medication at baseline then two weeks later.
The data are as follows (SBP in mmHg)
Subject Baseline
Week 2
Week 2
1
140
132
140
132
2
138 135
138 135
3
150 151
150 151
4
148 146
148 146
5
135 130
135 130
Consider testing the hypothesis that there was a mean reduction in
blood pressure? Give the Pvalue for the associated two sided T test.
blood pressure? Give the Pvalue for the associated two sided T test.
(Hint, consider that the observations are paired.)
 0.05
 0.10
 0.087
 0.043
Question 2)
A sample of 9 men yielded a sample average brain volume of 1,100cc and
a standard deviation of 30cc.
a standard deviation of 30cc.
What is the complete set of values of μ0 that a test of H0:μ=μ0 would
fail to reject the null hypothesis in a two sided 5% Students ttest?
fail to reject the null hypothesis in a two sided 5% Students ttest?
 1031 to 1169
 1077 to 1123
 1081 to 1119
 1080 to 1120
Question 3)
Researchers conducted a blind taste test of Coke versus Pepsi.
Each of four people was asked which of two blinded drinks given in
random order that they preferred.
random order that they preferred.
The data was such that 3 of the 4 people chose Coke.
Assuming that this sample is representative, report a Pvalue for a
test of the hypothesis that Coke is preferred to Pepsi using a one sided exact test.
test of the hypothesis that Coke is preferred to Pepsi using a one sided exact test.
 0.62
 0.005
 0.10
 0.31
Question 4)
Infection rates at a hospital above 1 infection per 100 person days at
risk are believed to be too high and are used as a benchmark.
risk are believed to be too high and are used as a benchmark.
A hospital that had previously been above the benchmark recently had 10
infections over the last 1,787 person days at risk.
infections over the last 1,787 person days at risk.
About what is the one sided Pvalue for the relevant test of whether
the hospital is *below* the standard?
the hospital is *below* the standard?
 0.11
 0.03
 0.52
 0.22
Question 5)
Suppose that 18 obese subjects were randomized, 9 each, to a new diet
pill and a placebo.
pill and a placebo.
Subjects’ body mass indices (BMIs) were measured at a baseline and
again after having received the treatment or placebo for four weeks.
again after having received the treatment or placebo for four weeks.
The average difference from followup to the baseline (followup –
baseline) was −3 kg/m2 for the treated group and 1 kg/m2 for the placebo group. The corresponding standard deviations of the
differences was 1.5 kg/m2 for the treatment group and 1.8 kg/m2 for the placebo group. Does the change in BMI appear to differ
between the treated and placebo groups?
baseline) was −3 kg/m2 for the treated group and 1 kg/m2 for the placebo group. The corresponding standard deviations of the
differences was 1.5 kg/m2 for the treatment group and 1.8 kg/m2 for the placebo group. Does the change in BMI appear to differ
between the treated and placebo groups?
Assuming normality of the underlying data and a common population
variance, give a pvalue for a two sided t test.
variance, give a pvalue for a two sided t test.
 Less than 0.01
 Less than 0.10 but larger than 0.05
 Larger than 0.10
 Less than 0.05, but larger than 0.01
Question 6)
Brain volumes for 9 men yielded a 90% confidence interval of 1,077 cc
to 1,123 cc.
to 1,123 cc.
Would you reject in a two sided 5% hypothesis test of H0:μ=1,078?
 #No you wouldn’t reject.
 #Yes you would reject.
 #It’s impossible to tell.
 #Where does Brian come up with these questions?
Question 7)
Researchers would like to conduct a study of 100 healthy adults to
detect a four year mean brain volume loss of .01 mm3.
detect a four year mean brain volume loss of .01 mm3.
Assume that the standard deviation of four year volume loss in this
population is .04 mm3.
population is .04 mm3.
About what would be the power of the study for a 5% one sided test
versus a null hypothesis of no volume loss?
versus a null hypothesis of no volume loss?
 0.60
 0.80
 0.70
 0.50
Question 8)
Researchers would like to conduct a study of n healthy adults to detect
a four year mean brain volume loss of .01 mm3.
a four year mean brain volume loss of .01 mm3.
Assume that the standard deviation of four year volume loss in this
population is .04 mm3.
population is .04 mm3.
About what would be the value of n needded for 90% power of type one
error rate of 5% one sided test versus a null hypothesis of no volume loss?
error rate of 5% one sided test versus a null hypothesis of no volume loss?
 180
 160
 120
 140
Question 9)
As you increase the type one error rate, α, what happens to power?
 #It’s impossible to tell given the information in the problem.
 #You will get smaller power.
 #No, for real, where does Brian come up with these problems?
 #You will get larger power.