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 σ (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 + σ Φ −1 (1/( k + 1)) is the value of a that minimizes the expected loss, where Φ −1 is 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?

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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
BuyFind

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

Solutions

Chapter 4, Problem 110SE
Textbook Problem

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 σ (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) = ta if t > a and = k(at) if ta (k > 1 is suggested to incorporate the idea that over-assessment is more serious than under-assessment).

a. Show that a* = µ + σΦ−1(1/(k + 1)) is the value of a that minimizes the expected loss, where Φ−1 is 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?

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Chapter 4 Solutions

Probability and Statistics for Engineering and the Sciences
Ch. 4.2 - Let X denote the amount of time a book on two-hour...Ch. 4.2 - The cdf for X (= measurement error) of Exercise 3...Ch. 4.2 - Example 4.5 introduced the concept of time headway...Ch. 4.2 - The article Modeling Sediment and Water Column...Ch. 4.2 - Let X denote the amount of space occupied by an...Ch. 4.2 - The article A Model of Pedestrians Waiting Times...Ch. 4.2 - Let X have a uniform distribution on the interval...Ch. 4.2 - Let X denote the voltage at the output of a...Ch. 4.2 - Let X be a continuous rv with cdf...Ch. 4.2 - Consider the pdf for total waiting time Y for two...Ch. 4.2 - An ecologist wishes to mark off a circular...Ch. 4.2 - The weekly demand for propane gas (in 1000s of...Ch. 4.2 - If the temperature at which a certain compound...Ch. 4.2 - Let X have the Pareto pdf f(x;k,)={kkxk+1x0x...Ch. 4.2 - Let X be the temperature in C at which a certain...Ch. 4.2 - Let X be the total medical expenses (in 1000s of...Ch. 4.2 - When a dart is thrown at a circular target,...Ch. 4.3 - Let Z be a standard normal random variable and...Ch. 4.3 - In each case, determine the value of the constant...Ch. 4.3 - Find the following percentiles for the standard...Ch. 4.3 - Determine z for the following values of : a. =...Ch. 4.3 - Suppose the force acting on a column that helps to...Ch. 4.3 - Mopeds (small motorcycles with an engine capacity...Ch. 4.3 - The article Reliability of Domestic-Waste Biofilm...Ch. 4.3 - In a road-paving process, asphalt mix is delivered...Ch. 4.3 - Spray drift is a constant concern for pesticide...Ch. 4.3 - Suppose that blood chloride concentration (mmol/L)...Ch. 4.3 - There are two machines available for cutting corks...Ch. 4.3 - The defect length of a corrosion defect in a...Ch. 4.3 - The article Monte Carlo SimulationTool for Better...Ch. 4.3 - The automatic opening device of a military cargo...Ch. 4.3 - The temperature reading from a thermocouple placed...Ch. 4.3 - Vehicle speed on a particular bridge in China can...Ch. 4.3 - If bolt thread length is normally distributed,...Ch. 4.3 - A machine that produces ball bearings has...Ch. 4.3 - The Rockwell hardness of a metal is determined by...Ch. 4.3 - The weight distribution of parcels sent in a...Ch. 4.3 - Suppose Appendix Table A.3 contained (z) only for...Ch. 4.3 - Consider babies born in the normal range of 3743...Ch. 4.3 - In response to concerns about nutritional contents...Ch. 4.3 - Chebyshevs inequality, (see Exercise 44, Chapter...Ch. 4.3 - Let X denote the number of flaws along a 100-m...Ch. 4.3 - Let X have a binomial distribution with parameters...Ch. 4.3 - Suppose that 10% of all steel shafts produced by a...Ch. 4.3 - Suppose only 75% of all drivers in a certain state...Ch. 4.3 - Show that the relationship between a general...Ch. 4.3 - a. Show that if X has a normal distribution with...Ch. 4.3 - There is no nice formula for the standard normal...Ch. 4.4 - Let X = the time between two successive arrivals...Ch. 4.4 - Let X denote the distance (m) that an animal moves...Ch. 4.4 - Data collected at Toronto Pearson International...Ch. 4.4 - The article Microwave Observations of Daily...Ch. 4.4 - A consumer is trying to decide between two...Ch. 4.4 - Evaluate the following: a. (6) b. (5/2) c. F(4; 5)...Ch. 4.4 - Let X denote the data transfer time (ms) in a grid...Ch. 4.4 - The two-parameter gamma distribution can be...Ch. 4.4 - Suppose that when a transistor of a certain type...Ch. 4.4 - The special case of the gamma distribution in...Ch. 4.4 - A system consists of five identical components...Ch. 4.4 - If X has an exponential distribution with...Ch. 4.4 - a. The event {X2 y} is equivalent to what event...Ch. 4.5 - The lifetime X (in hundreds of hours) of a certain...Ch. 4.5 - The authors of the article A Probabilistic...Ch. 4.5 - Once an individual has been infected with a...Ch. 4.5 - Let X have a Weibull distribution with the pdf...Ch. 4.5 - The article The Statistics of Phytotoxic Air...Ch. 4.5 - The authors of the article from which the data in...Ch. 4.5 - The article On Assessing the Accuracy of Offshore...Ch. 4.5 - Nonpoint source loads are chemical masses that...Ch. 4.5 - a. 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Suppose the lifetime X of a component, when...Ch. 4 - In Exercises 117 and 118, as well as many other...Ch. 4 - Based on data from a dart-throwing experiment, the...Ch. 4 - The article Three Sisters Give Birth on the Same...Ch. 4 - Let X denote the lifetime of a component, with f...Ch. 4 - Let U have a uniform distribution on the interval...Ch. 4 - Consider an rv X with mean and standard deviation...Ch. 4 - A function g(x) is convex if the chord connecting...Ch. 4 - Let X have a Weibull distribution with parameters ...Ch. 4 - An individuals credit score is a number calculated...Ch. 4 - Let V denote rainfall volume and W denote runoff...

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