Problem 1, Ear infections [3 points] A new vaccine was recently tested to see if it could prevent the painful and recurrent ear infections that many infants suffer from. The Lancet, a medical journal, reported a study in which babies about a year old were randomly divided into two groups. One group received vaccinations; the other did not. During the following year, only 325 of 2450 vaccinated children had ear infections, compared to 504 of 2457 unvaccinated children in the control group. (a) Are the conditions for inference satisfied? (b) Find a 95% confidence interval for the difference in rates of ear infection. (c) Use your confidence interval to explain whether you think the vaccine is effective. Problem 2, Pain [6 points] Researchers comparing the effectiveness of two pain medications randomly selected a group of patients who had been complaining of a certain kind of joint pain. They randomly divided these people into two groups, then administered the pain killers. Of the 115 people in the group who received medication A, 87 said this pain reliever was effective. Of the 111 people in the other group, 68 reported that pain reliever B was effective. (a) Write a 95% confidence interval for the percent of people who may get relief from this kind of joint pain by using medication A. Interpret your interval. (b) Write a 95% confidence interval for the percent of people who may get relief by using medication B. Interpret your interval. (c) Do the intervals for A and B overlap? What do you think this means about the comparative effectiveness of these medications? (d) Find a 95% confidence interval for the difference in the proportions of people who may find these medications effective. Interpret your interval. (e) Does this interval contain zero? What does that mean? (f) Why do the results in parts c and e seem contradictory? If we want to compare the effectiveness of these two pain relievers, which is the correct approach? Why? Problem 3, Running heats London [3 points] In Olympic running events, preliminary heats are determined by random draw, so we should expect the abilities of runners in the various heats to be about the same, on average. Here are the times (in seconds) for the 400-m women's run in the 2012 Olympics in London for preliminary heats 2 and 5. Is there any evidence that the mean time to finish is different for randomized heats? Explain. Be sure to include a discussion of assumptions and conditions for your analysis. (Note: One runner in heat 2 did not finish and one runner in heat 5 did not start.)

Country Name Time Heat BOT Amantle Montsho 50.4 JAM Christine Day 51.05 2 GBR Shana Cox 52.01 2 GUY Aliann Pompey 52.1 ANT Afia Charles 54.25 NCA Ingrid Narvaez 59.55 BAH Shaunae Miller DNF RUS Antonina Krivoshapka 50.75 JOG G G O O N N N NI UKR Alina Lohvynenko 52.08 GBR Lee Mcconnell 52.23 SWE Moa Hjelmer 52.86 MAW Ambwene Simukonda 54.2 FIJ Danielle Alakija 56.77 GRN Kanika Beckles DNS