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 web-based application Headspace for three weeks or completing a web-based 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 two-way 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 one-sided or two-sided? Why?

(e) Use the Two-way Tables applet to enter the data.

·        Check the Enter table box. Enter the appropriate counts, as well as short (one-word) 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 simulation-based 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 simulation-based p-value and press Count.

·        Include a screen capture of your null distribution, with the p-value displayed.

(g) Use the applet to find the “exact p-value”

·        Check the Show Fisher’s Exact Test p-value box.

Summarize the conclusion you would draw about this research question, in context, based on the exact p-value.

(h) Is a two-sample z-test (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 theory-based p-value.  Based on this comparison, do you consider the approximation reasonable? Suggest a strategy for improving this approximation of the p-value 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 p-value estimate.

(j) Determine and interpret (in context) the theory-based two-proportion 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 cause-and 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 flu-like symptoms by December 17, compared to 69 of the 402 who had not received the vaccine.


Not vaccinated



Developed symptoms




No symptoms








(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 long-run 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 pull-down menu to select the Relative Risk.

·        Press Draw Samples

·        Estimate the p-value.

·        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 p-value.  How has the p-value 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 p-value to the simulated p-value?

(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

riskratio(69, 149, 402, 1009, conf.level=0.95,
Explain the 4 values being input (see the table output by the function too)


Enter the two-way 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 p-value and/or confidence interval?

(k) Are you willing to draw a cause-and effect conclusion from this study?  Explain why or why not.

(l) To what population are you willing to generalize these results?  Justify your choice.