Chapter 6 Exercises
1. The physician’s health study (phase I) involved over 22,000 US physician’s being
randomly assigned to received aspirin or placebo. Physicians were followed and
kept track of for four years, keeping record of which physicians had heart attacks in
order to see if aspirin reduced the risk of heart attacks.
a) Identify the research question and explanatory/response variables.
b) State the null and alternative hypotheses for the related test of significance.
c) Ultimately, 189 out of 11,034 placebo taking physicians had a heart attack and
104 out of 11,037 aspirin taking physicians had a heart attack. Find the
difference in rates of heart attack.
d) The p-value for the hypothesis test on the difference in rates of heart attack was
reported as 0.0000057. What is your conclusion for this test of significance?
e) Why do you think the p-value is so small?
f) The original study was scheduled to last 10 years (1985-1995). Instead,
however the study was halted after 4 years (1989). Give some of the
moral/ethical and statistical reasons why you think the study was stopped after
only 4 years.
g) How do you think the p-value would have changed had the alternative hypothesis
been two-sided instead of one-sided? Would it have impacted your conclusions?
2. A study of the comparison of the proportion of boys born to smoking parents to that
of nonsmoking parents was reported on April 20, 2002 by The Lancet, a British
medical journal. The results of the article showed that couples who smoke around
the time of conception are less likely to produce boys than those who do not. One of
the statistics reported that out of 565 births where both parents smoked more than a
pack a day, 255 were boys. Another statistic reported that out of 3602 births where
both parents did not smoke, 1975 were boys. A p-value comparing the difference in
proportions between these two groups is 0.000017.
a) What proportion of births resulted in a boy when both parents smoked more than
a pack a day? What proportion of births resulted in a boy when both parents did
not smoke?
b) State the null and alternative hypotheses and give your conclusion based on the
p-value.
c) Identify the explanatory and response variables.
d) Is this study an observational study or an experiment? How does this impact
your conclusions?
1
Chapter 3: Comparing Two Proportions
3. Covering a wart with a piece of duct tape may be as
effective in getting rid of it as liquid nitrogen freezing,
according to an article in the October 2002 issue of the
Archives of Pediatrics & Adolescent
Medicine. Researchers from Madigan Army Medical
Center in Tacoma, Washington, studied 51 patients ages 3
to 22 with common warts. Twenty-six patients were treated
with duct tape and 25 were treated with liquid nitrogen, or
cryotherapy.
Patients in the tape group, or their parents, were told to
leave the tape in place for six days, and to replace it if it fell
off. After six days, they were told to remove the tape, soak
the area in water, and file the wart with an emery board or
pumice stone. After 12 hours without the duct tape, they
were told to put a new piece on the wart, and continue the
cycle for two months or until the wart was gone. Patients
in the cryotherapy group received a standard application of
liquid nitrogen on the wart for 10 seconds. Patients, or their
parents, were told to return to the clinic every two to three
weeks to repeat the freeze for a maximum of six
treatments or until the wart was gone.
The researchers found that the duct tape treatment
completely removed warts in 22 of 26 patients, while the
liquid nitrogen treatment removed warts in 15 of 25
patients. From these two sample proportions can we
conclude that treating a wart with duct tape is better than
cryotherapy?
a) State the research question.
b) State the null and alternative hypotheses for this study.
c) Identify the explanatory and response variables.
d) Is this study an experiment or observational study?
Why?
e) Find the percent of successful wart removal in each of
the two treatment groups.
f)
Use your answer to (e) to find the difference in
proportions (duct tape minus liquid nitrogen).
g) We used fathom to do 1000 scramblings and found the
difference in proportions each time. Use your answer
to (f) and the table on the right to find the p-value.
2
Measures f rom Scrambled Collection 1
propDuct
propN
diffProps
1
0.961538
0.48
0.481538
2
0.923077
0.52
0.403077
3
0.923077
0.52
0.403077
4
0.884615
0.56
0.324615
5
0.884615
0.56
0.324615
6
0.884615
0.56
0.324615
7
0.884615
0.56
0.324615
8
0.884615
0.56
0.324615
9
0.846154
0.6
0.246154
10
0.846154
0.6
0.246154
11
0.846154
0.6
0.246154
12
0.846154
0.6
0.246154
13
0.846154
0.6
0.246154
14
0.846154
0.6
0.246154
15
0.846154
0.6
0.246154
16
0.846154
0.6
0.246154
17
0.846154
0.6
0.246154
18
0.846154
0.6
0.246154
19
0.846154
0.6
0.246154
20
0.846154
0.6
0.246154
21
0.846154
0.6
0.246154
22
0.846154
0.6
0.246154
23
0.846154
0.6
0.246154
24
0.846154
0.6
0.246154
25
0.846154
0.6
0.246154
26
0.846154
0.6
0.246154
27
0.846154
0.6
0.246154
28
0.846154
0.6
0.246154
29
0.846154
0.6
0.246154
30
0.846154
0.6
0.246154
31
0.846154
0.6
0.246154
32
0.846154
0.6
0.246154
33
0.846154
0.6
0.246154
34
0.846154
0.6
0.246154
35
0.846154
0.6
0.246154
36
0.846154
0.6
0.246154
37
0.846154
0.6
0.246154
38
0.846154
0.6
0.246154
39
0.846154
0.6
0.246154
40
0.846154
0.6
0.246154
41
0.846154
0.6
0.246154
42
0.846154
0.6
0.246154
43
0.846154
0.6
0.246154
44
0.846154
0.6
0.246154
45
0.846154
0.6
0.246154
46
0.846154
0.6
0.246154
47
0.846154
0.6
0.246154
48
0.807692
0.64
0.167692
49
0.807692
0.64
0.167692
50
0.807692
0.64
0.167692
51
0.807692
0.64
0.167692
h) Based on your p-value what would be your conclusion for the study?
<ne w >
Chapter 3: Comparing Two Proportions
3
4. If a plant is attacked by one organism, does this induce resistance to subsequent
attacks by a different organism? To test this research question, individually potted
cotton plants were randomly allocated to two groups. One group was infested by
spider mites and the other wasn’t. After two weeks, the spider mites were removed
and both groups were inoculated with Verticillium, a fungus that causes Wilt disease.
Use the data set mites.ftm to answer the research question.
a) State your null and alternative hypotheses. The research question does have a
direction to it. However, we are unsure if increased resistance will be the
outcome. It could be that the plant, in a weakened state, is more susceptible to
attacks by a different organism. As is often the case in research, we hope to
prove a certain directional outcome, but frame our alternative in either direction to
account for uncertainty in our outcome and to be more conservative in our
testing.
b) Create a 2x2 table in Fathom.
What is your explanatory variable? (Remember this is your column variable.)
What is your response variable? (Remember this is your row variable.)
c) What proportion of non-mite infested plants did NOT develop Wilt disease?
What proportion of mite infested plants did NOT develop Wilt disease?
Does there appear to be an association between infestation by mites and Wilt
disease? State this in words and support it with descriptive statistics (row
proportions). Is it in the direction of our research question?
d) The measure we will use to compare the two groups is: the proportion of plants
without Wilt disease that were NOT initially infested with mites – the proportion of
plants without Wilt disease that were initially infested with mites. Calculate this
measure for the observed data.
e) Use Fathom to calculate the p-value for the test by scrambling the explanatory
variable 1000 times, each time calculating and recording the measure. You will
need to create the measure for the difference in proportions before you scramble.
State your p-value.
f) State your conclusions. Don’t forget to talk about what population inferences can
be made to and whether or not cause and effect conclusions can be made.
5. The following table contains information from a study of 200 randomly sampled
Intensive Care Unit (ICU) patients on patient survival according to their race.
Survival Status
Lived
Died
White
138
37
Race
Non-white
22
3
a) What percent of white patients died in ICUs? What percent of non-white patients
died in ICUs?
b) Based on your descriptive statistics do you believe there is evidence of an
association between race and survival status?
c) Carry out the appropriate test following all steps for a test of significance using
the data set ICU Survival and Race.ftm to see if there is an association between
race and survival rates in the ICU.