A public health researcher is investigating the effectiveness of fluoride in preventing cavities. Two communities are studied, one with a fluoridiated water supply and the second with non-fluoridated water. A random sample of n1 = 400 and n2 = 900 residents are obtained from these two communities, respectively. It was found that y1 = 348 residents in the first community and y2 = 819 residents in the second community had cavities. Consider modeling these counts by two binomial distributions with the probability of success (presence of cavities) as 71 and 72 for communities with fluoridated and non-fluoridated water supplies, respectively. (i.e. consider the random variables Y1 ~ Bin(n1, 71) and Y2 ~ Bin(n2, 72).) a. Let P1 = Y11 and P2 = Y22 be random variables denoting the proportion of sampled residents with cavities in each population. Approximate the sampling distributions of P1 and P2. b. Based on your answer from (a), approximate the sampling distribution for the difference in sample proportions, D = P1 - P2. (Hint: think of this as a sum of normal random variables. ) c. Suppose we use the statistic D as an estimator for the true difference 71 - 72. Calculate the bias and standard error of this estimator. (Note, these may be a function of 71, 72, n1, and n2.) d. Assume that 71 = 12 = 0.9. Then, using your approximation in (b), calculate P(|D| > d), where d = y11 - y22 is the observed difference found by the researcher. e. Based on your answer in part (d), do you think the assumption that 71 = 12 = 0.9 is contradicted by the data? Briefly explain why/why not. (Note: there is no right/wrong answer here, but you need to give some probabilistic reasoning. This reasoning may be similar to the logic behind hypothesis testing, even though you're not performing a formal statistical test in this problem.)