3. Question You invented a new lateral-flow test for asymptomatic Covid-19 detection, where saliva is entered into a test-tube and then the result is shown directly on the device. However, to save on costly chemicals you designed the tests such that it always reports a negative test result. Assume that the incidence rate is 5 per 1000 and that your test is used for detection of asymptomatic cases (without symptoms). (a) Calculate the probability that 100 randomly determined volunteers receive a correct test result by using the AND-rule and the OR-rule. Can your test be qualified as a diagnostic test? (b) The health authorities are investigating the performance of your test. Government guidelines require a specificity (conditional probability to re- ceive a negative test result given that the test-taker is not infected with COVID19) of at least 97% and a sensitivity (conditional probability to receive a positive test result given that the test-taker is infected with COVID-19) of at least 80%. Use Bayes' rule to check whether your test satisfies both requirements. (c) The health authority becomes highly suspicious of your test. They suspect that with 10% probability your test is always reporting negative results and with 90% probability your test is always correct, (i.e. it has a speci- ficity and sensitivity of 100%). The government authority is informed about the test result of one patient but cannot verify whether the respec- tive patient was actually infected or not. Represent this situation as a Bayesian updating problem by identifying the prior and calculating the posterior of the health authorities. (d) Assume now that the incidence rate increased to a rate of 5 per 10. Does this change your results from part b)? Discuss briefly the concept of base- rate neglect in this context and justify whether it is applicable in this case or not. Does this change affect Bayesian updating addressed in part c)? (e) You decide to tweak your test in such a way that in 0.5% of cases a positive result is shown. Repeat your calculations from part a) assuming the initial incidence rate of 5 per 1000. Does this tweak improve the predictive power of your test?