Stat 342 Homework 4 Spring 2016 Due Friday, February 19th at the beginning of class Make sure to show all your work. Points are given for both, the correct answer and a detailed solution to the correct answer. 1. Let Y1 , Y2 , . . . , Yn be independent and identically distributed with the following probability density function f (y) = 4(1 y)3 0 < y < 1 0 otherwise (a) Find the probability density function of Y(1) = min(Y1 , . . . , Yn ). (b) Find the probability that Y(1) is less than 0.1. (c) Find the expected value of Y(1) . (Hint: beta distribution). (d) Find the variance of Y(1) . (Hint: beta distribution). 2. Let Y1 , Y2 , . . . , Yn be independent and identically distributed random variables from a Uniform distribution with minimum value 1 = 0 and maximum value 2 = . (a) Find the probability density function of Y(n) = max(Y1 , . . . , Yn ). (b) Find the expected value of Y(n) . (c) Find the variance of Y(n) . (d) Use your answer to the previous three parts to describe in words what happens to the distribution of Y(n) as the sample size n . 3. In many states a motorist is legally drunk or driving under inuence (DUI), if his or her blood alcohol concentration, Y , is 0.10% or higher. When a suspected DUI oender is pulled over, police often request a sobriety test. Although the breath analyzers used for that purpose are remarkably precise, the machines exhibit a certain amount of measurement error. Because of that variability, the possibility exists that a driver's true blood alcohol concentration may be under 0.10% even though the analyzer gives a reading over 0.10%. Experience shows that repeated breath analyzer measurements taken on the same individual produce a distribution of responses that can be described by a normal pdf with equal to the person's true blood alcohol concentration and = 0.004%. (a) Suppose a driver is stopped at a roadblock on his way home from a party. Having celebrated a bit more than he should have, he has a true blood alcohol concentration of 0.095%, just barely below the legal limit. If he takes a breath analyzer test, what are the chances that he will be incorrectly booked on a DUI charge? (b) Suppose the driver's blood alcohol concentration is actually 0.11% rather than 0.095%. What is the probability that the breath analyzer will make an error in his favor and indicate that he is not legally drunk? 1 (c) Suppose the police oer the driver a choice - either take the sobriety test once or take it twice and average the reading? Which option should a \"0.095%\" driver take? Which option should a \"0.11%\" driver take? Explain using a complete argument that is statistically sound, that is your answer should include some numerical values and not just words :-)! 4. A population has mean =(25 points)standard deviation = 2. Repeated samples each one of 1.26 and Problem 4 size n = 40 are Part A: from this population. It is claimed that the histogram below represents the selected A total (T ) has mean observations. Clearly explain = 2. Repeated samples each one distribution of thepopulation of these 40 = 1.25 and standard deviation if there is anything wrong withof size n = this histogram. are selected from this population. It is claimed that the histogram below represents the distribution of t 0.04 0.00 0.02 Prob. density of T 0.06 total (T ) of these 40 observations. Clearly explain if there is anything wrong with this histogram. 29 34 38 42 46 50 53 56 59 62 65 68 T Part B: Let X1 , X2 , . . . , X19 be a random sample from N (0, ), and let X, S 2 represent the sample mean and sam variance of this sample. Answer the following questions: 2 a. For what value of c the expression c X2 follows the F distribution? What are the degrees of freedom S b. Using only your statistical tables (please no calculator!) nd the 80th percentile of the distribution 2 c X2 . Show all your work. S 2