In the endemic phase of a virus spread, testers are employed to carry out a random testing in a community of 400,000 people. There are a total of 8 testers, each tasked to test 20 persons per day randomly selected from the community, over a duration of 90 days. There is no repeated testing of any individual. The number of persons who tested positive on a daily basis is submitted by each tester and provided in the file "test_results'.

(a) Consolidate the data from all 8 testers. Then, plot the distribution of the number of positive cases per day (per tester) over the entire dataset. (The total number of samples in your entire distribution will be 8?90 = 720 in this case.)

(b) Your team leader suspects that the number of positive cases per day follows a binomial distribution. By considering the conditions for such a model, demonstrate why it might be suitable.

(c) By comparing the PMF of a binomial model with the chart obtained in part (a), determine the probability of contracting the virus for any individual by trial and error, up to ONLY 1 decimal place. Then, plot the distribution of the predicted number of positive cases per day using the binomial model in the same graph with the actual distribution (shown in part (a)) and discuss the accuracy of the model.

(d) The team leader has been informed that one of the 8 testers received a faulty batch of test kits that was used throughout the 90 days. Describe and execute your strategy to discover this tester. What is wrong with the test-kits used by this tester? Remove the faulty test results and graph the comparison between the predicted and actual cases again. State your observation.

(e) A team member suggests that there is a more than 10% chance that at least 8 persons out of 20 tested will be positive cases. Using your binomial model, explain the reasons to support or reject this claim.

(f) For only tester 7, calculate the mean, standard deviation, median, third quartile and 90th percentile of his testing results.

(g) Plot the PMF and CDF of the geometric distribution from the Bernoulli trial of the virus test for each tester per day. Hence, compute the probability that a tester obtains the first positive case within the 20 tests allocated in a single day. Why is this probability not 1?

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