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 standard deviation θ (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 a* = μ + σɸ-1(1 / (k + 1)) is the value of a that minimizes the expected loss, where ɸ-1 is the inverse function of the standard normal cdf.

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