Solved: Let t = the amount of sales tax a retailer owes

Chapter 4, Problem 110E

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Let t = the amount of sales tax a retailer owes the government for a certain period. The article “Statistical Sampling in Tax Audits” (Statistics and the Law, 2008: 320–343) proposes modeling the uncertainty in t by regarding it as a normally distributed random variable with mean value \(\mu\) and standard deviation \(\sigma\) (in the article, these two parameters are estimated from the results of a tax audit involving n sampled transactions). If a represents the amount the  retailer is assessed, then an under-assessment results if t > a and an over-assessment results if a > t. The proposed penalty (i.e., loss) function for over- or under-assessment is L(a, t) = t > a if t > a and = k(a - t) if \(t \leq a\) (k > 1 is suggested to incorporate the idea that over-assessment is  more serious than under-assessment).

a. Show that \(a^* - \mu +\sigma \Phi^{-1}(1/k+1))\) it the value of a that minimizes the expected loss, where \(\Phi^{-1}\) is the inverse function of the standard normal cdf.

b. If k = 2 (suggested in the article), \(\mu\) = $100,000 and \(\sigma\) = $10,000, what is the optimal value of a, and what is the resulting probability of over-assessment?