Let t = the amount of sales tax a retailer owes the governmentfor a certain period. The

Chapter 4, Problem 110

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Let t = the amount of sales tax a retailer owes the governmentfor a certain period. The article StatisticalSampling in Tax Audits (Statistics and the Law, 2008:320343) proposes modeling the uncertainty in t byregarding it as a normally distributed random variablewith mean value m and standard deviation s (in the article,these two parameters are estimated from the results ofa tax audit involving n sampled transactions). If arepresents the amount the retailer is assessed, then anunder-assessment results if t . a and an over-assessmentresults if a . t. The proposed penalty (i.e., loss) functionfor over- or under-assessment is L(a, t) 5 t 2 a if t . aand 5 k(a 2 t) if t # a (k . 1 is suggested to incorporatethe idea that over-assessment is more serious thanunder-assessment).a. Show that a*5 m 1 sF21(1y(k 1 1)) is the value ofa that minimizes the expected loss, where F21 is theinverse function of the standard normal cdf.b. If k 5 2 (suggested in the article), m = $100,000, ands 5 $10,000, what is the optimal value of a, and whatis the resulting probability of over-assessment?