Problem 3 (21 pts). In an air pollution study, a random sample of 20 households were selected from each of two communities A and B. A reSpondent in each household was asked whether or not anyone in the household was bothered by air pollution. Here are the collected data: ommunity ' l-I- .o Inere y a1r p0 uion ' . 'es . 1 ota I I II A researcher wishes to know if people in the two communities are equally bothered by air pollution. (a) (2 pts) Write the hypotheses in words H : O H . 1 . (b) (5 pts) Suggest tWO methods to perform the test for H0 versus H1: Method I: Method II: For these two methods, method may be more reliable for this dataset ,because (one sentence) (c) (3 pts) If X and Y are independent , the expected cell count for the rst cell (i.e., the cell with count 4) is , and the difference between the expected cell count and the observed cell count for the first cell is (d) (5 pts) If the sum of the differences between all the observed cell counts and the corre- sponding expected cell counts is large , it suggests that the hypothesis is more likely to hold. To determine if the sum of the differences is large or not, we can compare the test statistic (write a formula ) to the 95% per centile of the Warameter values or degrees of m approximation is acceptable ), and we reject H a if (e) (2 pts} If there are three communities (instead of two), assuming a large sample, the null distribution is (specify parameter values or degrees of freedom as appro - priate ) . (f) (4pts) Suppose that we also wish to test if there are equal number of individuals who are bothered by and not bothered by air pollution, ignoring which community they are from. We decide to reject the null hypothesis of equal number if 25 or more individuals (out of the total 40) are not bothered by air pollution .The power of the test when in fact 60% individuals are not bothered by air pollution is given by (Show the key steps. No need to compute the final answer)