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

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 μ and σ standard deviation s (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 ≤ a (k >1) is suggested to incorporate the idea that over-assessment is more serious than under-assessment).

a. Show that is the value of that minimizes the expected loss, where ?^–1is the inverse function of the standard normal cdf.

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