STAT 67, Fall 2016 Homework 5 11-29-2016 1. Say you have n = 100 many Xi 's, where the Xi 's are independent and identically distributed Bernoulli random variables with p = 0.5 (E(X)=p and var(X)=p(1-p)). n P a. What distribution does Xi follow exactly? State the type of distribution and what the i=1 parameter is. b. Using the CLT, what approximate distribution does n 1P Xi n i=1 follow? State the type and the mean and variance. = c. What is the approximate probability that X n 1P Xi n i=1 is greater than 0.5? Write down the integral using proper notation and use empirical rule or pnorm function in R to obtain numeric answer. 2. In an experiment designed to measure the time necessary for an inspectors eyes to become used for n = 50 to the reduced amount of light necessary for inspection, the sample average time, X, inspectors was 6.32 seconds and the sample standard deviation, s, was 1.65 seconds. a. Using the central limit theorem, what approximate distribution does X s/ n follow (leave the unknown parameters in notation, such as and )? b. Compute an approximate 95 percent confidence interval for the true average adaptation time (the 95% multiplier is 2). 3. An airline company models the weight of each passengers luggage, Xi , as independent and identical random variables according to a normal distribution with mean 35 pounds and standard deviation of 5 pounds. a. Now say the airline converts from pounds to kilograms. Let Y be the weight of the luggage in kilograms, Yi = 0.45Xi . What distribution does Yi follow. b. Now say the airline relaxes the normal distribution specification. So each luggage weights Xi are i independent and identically distributed from some distribution with mean 35 pounds and standard deviation of 5 pounds. = What approximate distribution does X n 1P Xi n i=1 follow? State the type, mean, and variance. c. The airline is more interested in the total weight of all the luggages, rather than the mean of n P them. What approximate distribution does Xi follow? State the type, mean, and variance. i=1 d. Say a flight has 100 passengers. What is the approximate probability that the total weight of the luggages will be more than 3600 pounds? Write out the integral and compute value using empirical rule or pnorm in R. 4. let Xi be a random variable from a uniform distribution on the interval [0,b]. So the lower bound is known but not the upper bound. You observe n many independent and identically distributed n = 1 P Xi ? How would you use X to come Xi 's. What is the expected value of the sample mean X n i=1 up with an estimate of that the unknown parameter b is? 5. Let X1 and X2 be independent and identically distributed normal random variables with mean and standard deviation . a. What distribution does X1 follow? \u00112 b. What distribution does X1 follow? \u00112 \u0010 \u00112 \u0010 c. What distribution does X1 + X2 follow? \u0010