Stat 301  HW 6
Due midnight, Friday,
Feb. 24
Remember to put your name(s) inside each file and, if
submitting together, join a HW group first. Remember
to show your work/calculations/computer details and to integrate this into the
body of the solution.
1) Researchers
at Northwestern University explored whether meditation helps someone become a
more compassionate person (Lim,
Condo, & DeSteno, PLoS One, 2015).
To do this, they recruited 56 university students, all of whom reported little
to no prior experience with meditation. The students were randomly assigned to
one of two conditions: regularly completing a meditation session using the
webbased application Headspace for three weeks or completing a webbased
cognitive training program from Lumosity for three weeks (considered an “active
control group”). To test the subjects on their level of compassion they staged
a scenario using three actors. The research subjects would enter a common
waiting room where there were three chairs. Two male actors sat in two of the
chairs leaving one for the research subject. After the research subject was
sitting for one minute, a female actor came in playing the role of a person
suffering. She would walk in using crutches with some mild expressions of pain
and then would lean against the wall with a sigh of discomfort. The sitting male
actors were trained to ignore her. What did the research subjects do? It turned
out that 10 of the 27 from the meditation group got up and offered the
suffering woman their seat, whereas only 4 of the 29 in the active control
group did so.
(a) Identify the observational units, explanatory variable, and
response variable for this study. Which response variable outcome will you
consider “success”?
(b) Was this study observational or experimental? Explain how you are
deciding.
(c) Create a twoway table of these results, using the explanatory
variable for the columns.
(d) Define the parameter of interest in words and symbols and state
appropriate null and alternative hypotheses. Clarify what your hypotheses are
testing (which may not exactly match the research question?). Is your
alternative hypothesis onesided or twosided? Why?
(e) Use the Twoway Tables applet
to enter the data.
·
Check the Enter table box. Enter the appropriate counts, as well as
short (oneword) column and row names.
o Note: the applet allows you
to enter in the subtraction expression to find the number of “failures.”
o Be sure to press Use Table
when you are done.
·
Check the Show Table box and include a screen capture of the graph
(either a bar graph or mosaic plot) and the observed two way table.
·
Report the statistic, being sure to clarify the direction of
subtraction.
(f) Use the applet to carry out a simulationbased randomization test:
·
Check the Show Shuffle Options box.
·
Enter a large number of shuffles and press Shuffle.
·
Use the Count Samples box to find the simulationbased pvalue and
press Count.
·
Include a screen capture of your null distribution, with the pvalue
displayed.
(g) Use the applet to find the “exact pvalue”
·
Check the Show Fisher’s Exact Test pvalue box.
Summarize the conclusion you would draw about this research question,
in context, based on the exact pvalue.
(h) Is a twosample ztest (aka normal approximation to the hypergeometric distribution)
likely to be valid for this data? Explain how you are deciding.
(i) Check the Overlay normal distribution box and include a screen
capture of the null distribution with the theorybased pvalue. Based on this comparison, do you consider the
approximation reasonable? Suggest a strategy for improving this approximation
of the pvalue and roughly carry out this strategy by using your mouse to move
the red “count line.” Don’t worry about being too precise here, just explain
the process. Include a screen capture of your null distribution and new pvalue
estimate.
(j) Determine and interpret (in context) the theorybased
twoproportion 95% confidence interval for the difference in the probability of
“getting up.” Make sure your interpretation clearly defines the parameter and
the direction of the difference you find.
·
Use the “Wilson adjustment” (p. 176)
·
Check the box for 95% CI(s) for difference in proportions
(k) Are you willing to draw a causeand effect conclusion from this
study? Explain why or why not.
(l) To what population are you willing to generalize these
results? Justify your choice.
(m) Explain why the above analysis does not help you answer this
research question: After meditation, are individuals more likely to give up
their seat than to not give up their seat?
2) In January of 2004, the
Centers for Disease Control analyzed preliminary data on the effectiveness of a
flu vaccine given to workers at Children’s Hospital in Denver, Colorado. The hospital sent an anonymous survey to
approximately 3100 hospital workers, and 1866 responded. From these 1866 responses, 1818 were included
in the study, after some were eliminated for not responding to all of the
questions. Of the 1818 hospital workers
in the study, 1009 had opted to receive the vaccine before November 1, and an
additional 425 had opted to receive the vaccine on or after November 1, leaving
402 who opted not to receive the vaccine.
The 425 who received the vaccine on or after November 1 were also excluded,
because it was not clear that the vaccine would have had time to prove
effective by December 17, when the study results were compiled for
analysis. Of the 1009 who had been
vaccinated before November 1, 149 experienced flulike symptoms by December 17,
compared to 69 of the 402 who had not received the vaccine.

Not vaccinated 
Vaccinated 
Total 
Developed
symptoms 
69 
149 
218 
No
symptoms 
333 
860 
1193 
Total 
402 
1009 
1411 
(a) Calculate the difference in conditional
proportions developing symptoms ().
(b) Calculate the relative risk of
developing symptoms ().
(c) Complete this statement: There was a
____ % increase in the likelihood of symptoms for those who were not
vaccinated compared to those who were vaccinated.
(d) Let represent the “population” relative risk for
the longrun treatment probabilities. State
appropriate null and alternative hypotheses in terms of this parameter for
whether unvaccinated individuals are more likely to develop symptoms than
vaccinated people.
(e) Use the Two Population
Proportions applet
to simulate a null distribution with the relative risk as the statistic.
·
Enter
the “pooled” estimate of the population proportion with symptoms.
·
Enter
the two sample sizes.
·
Enter
a large number of samples.
·
Use
the Choose comparison pulldown menu to select the Relative Risk.
·
Press
Draw Samples
·
Estimate
the pvalue.
·
Include
a screen capture of your results, showing both the input information and the
output.
(f) Change the comparison to the ln
relative risk. Calculate the observed
value for this statistic. Use the new
null distribution to estimate the pvalue.
How has the pvalue changed?
(g) Check the Normal Approximation box.
Does the normal approximation appear valid here, comments on each of these:
·
Based
on the overlay of the blue distribution?
·
Based
on comparing the normal approximation pvalue to the simulated pvalue?
(h) Use the formula on p. 211 to
approximate the standard deviation for the ln relative risk. Show your work.
How does this theoretical SD compare to your simulation results? (Be very clear
what two numbers you are comparing.)
(i) Use R or JMP or your calculation to
calculate and interpret a 95% confidence interval for the relative risk.
R users can try install.packages("fmsb") library(fmsb) riskratio(69, 149, 402, 1009, conf.level=0.95, p.calc.by.independence=TRUE) Explain the 4 values being input (see the table output by the function too) 
JMP Enter the twoway table (see p.
171) and choose Analyze > Fit Y by X. Use the hot spot to select
Relative Risk. Check the box to calculate all combinations and then
pick one to interpret. 
(j) What conclusion do you draw about the research questions from the
pvalue and/or confidence interval?
(k) Are you willing to draw a causeand effect conclusion from this
study? Explain why or why not.
(l) To what population are you willing to generalize these
results? Justify your choice.