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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 underassessment results if $t>a$ and an overassessment if $a>t .$ We can express this in terms of a loss function, a function that shows zero loss if $t=a$ but increases as the gap between $t$ and $a$ increases. The proposed loss function 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 $a *=\mu+\sigma \Phi^{-1}(1 /(k+1))$ is 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-assent?

(b) $95,600, .3300

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

Temple University

Missouri State University

University of North Carolina at Chapel Hill

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So 97 has quite a few parts. So for a you just have to pull the information out of a problem. So X bar is the sample mean? I was 23.6 hours sigma, which is the population standard deviation with seven hours and and how many are sample was 100. Hi. B wants us to define our variable X and X bar. So are variable X is actually the length of time it took to fill out your taxes. An exit bar was the mean of those 100 people. So it's our sample. Me see which distribution? Since we had over 30 samples, we have the normal distribution 23.6 and something of the population standard deviation to seven over the square root of behind D wants us to come up with the confidence interval. So 23.6 plus or minus. Do you need the confidence interval for 90% at 1.645 and then times seven over the square root of 100. So that gives us 23.6 plus or minus one point 0.515 And so we do this attraction in the addition, we get 22 0.4485 and the upper limit is 24 0.75 15 So we are 90% sure that the true me length of time. The Finnish one's taxes is somewhere between approximately 22.45 hours in 24.75 hours. The the firm wish to increase its level of confidence and keep the air bound the same by taking another survey what changes would be made so you wouldn't need more samples so and would have to be higher for F at the firm did. Another survey kept the air bound the same. It only surveyed 49 people, so if they had less samples, they would have a wider interval. And G wants us to really come up with the sample size. So since now we want to be at least 96% confident. We know our end is gonna be larger. So what we end up with since we want to be within one hour and 96% confident you would have one equals 2.54 critical value. What times seven over the square Reader then. So if you square it, multiply both sides by n move Everything around you end up with n equals 2.54 square times seven squared over one squared. That would give you 206.7 people. This always rounds up, so we would need 207.

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