A recent study (Ackerman, Griskevicius, and Li, 2011) examined expressions of commitment between two partners in a committed romantic relationship. One aspect of the study involved 47 heterosexual couples who were asked about which person was the first to say "I love you." For 7 of those couples, the two people disagreed about the answer to this question. But both people agreed for the other 40 couples, so those 40 responses were included in the analysis. Is one sex more likely to say "I love you" first? a) Define the parameter of interest in words and indicate the symbol used to represent it. b) State the null and alternative hypotheses, both in symbols and in words. (Hint: your alternative hypothesis should be consistent with the research question as stated above, and not your own personal conjecture.) c) For 28 of the 40 couples in the sample, the man said "I love you" before the woman did. Compute by hand the z statistic, sketch a picture depicting the p- value, and provide an educated estimate of the p-value. Explain the reasoning behind your estimate. (Of course, you can use the applet to check your z statistic and p-value, but make sure you know how to get a ballpark estimate of the p-value based on the z statistic without the use of software.) d) Use the usual applet to compute the estimate the p-value via simulation. Is the p-value based on the normal approximation relatively close to the simulated value? Why should this have been expected? (Hint: What are the conditions for the normal approximation to be valid? Were they satisfied here?) e) Write a clearly worded sentence stating your conclusion. (Hint: remember how you stated the alternative hypothesis.) f) Would you conclude that 0.5 is a plausible value of the parameter, using a strict significance level of a=0.05 as the criterion for rejection/plausibility? Explain (briefly) how you decide. g) Repeat the previous part with a=0.01.