1.) Let ? be a deck of cards. Define a random variable X such that for any card c in the deck the number value on the card for all numbered cards X (c) = 10 for any face cards (jack, queen, king) 11 for any ace What is E(X )? 2.) For a continuous random variable X, where c is any one isolated number what is: P(X = c) Hint: For continuous random variables we always calculate things like: 3.) Suppose annual snow fall is normally distributed with a = 54 inches and o = 6. What is the probability that within a given year there will be snowfall totalling between 48 inches and 56 inches? 4.) The central limit theorem says as we increase the number of samplings n, the variance of the sampling distribution will 5.) Diesel Dave has modified his massive truck to combat global cooling, and to roll coal on his neighbor Environmentalist Ed. With each blast of diesel exhaust fumes, Diesel Dave releases sulfur dioxide in a giant black cloud. Every morning as Diesel Dave is leaving home he rolls coal into Environmentalist Ed's yard. Not surprisingly, Environmentalist Ed is diligently collecting data on the air quality in his yard. Suppose Diesel Dave releases sulfur dioxide in a normally distributed manner with a mean of 2800 ppm and a standard deviation of 120 ppm. What is the probability that on a random morning he will release sulfur dioxide greater than 3000 ppm?6.) Suppose in order to sue Diesel Dave, Environmentalist Ed needs the daily average level of sulfur dioxide emissions to exceed 3000 ppm. However, he can choose to take his mean measurement over 2, 3, or 15 days. Which number of days would maximize Environmentalist Ed's chances of suing Diesel Dave? (Hint: What does the CLT say about the variance of the sampling distribution as n changes?) 7.) Miller Beer needs its beer to be somewhat bitter, but not too bitter, in order to maximize sales revenue from Wisconsin Bros. The standard way of measuring bitterness is the number of IBUs or International Bitterness Units. A batch of Old Milwaukee beer has on average 24 IBUs with a standard deviation of 3.1 IBUs, moreover IBUs are distributed normally across beer batches. If the batch of beer has more than 28 IBUs Wisconsin Bros declare the beer "liberal, coastal elite beer" and refuse to drink it. If the beer has fewer than 18 IBUs Wisconsin Bros declare the beer "weak" and also refuse to drink it. What is the probability that a randomly drawn batch of beer will be undrinkable to Wisconsin Bros? 8.) Suppose that Miller Beer samples people's perceptions of the quality of Old Milwaukee beer. Respondents are asked to evaluate the beer on a scale of 1 to 10. The modal response is a 9 amongst sampled Wisconsin Bros, while the mean is 7.7 with a standard deviation of 2. Suppose you conduct 27 samplings to generate your point estimate of 7.7 as the mean evaluation of Old Milwaukee beer. If you want to use the central limit theorem to evaluate the likelihood that the next sampling will generate a mean evaluation greater than 8.0 which of the three assumptions is clearly violated and why (Hint: If you don't think it's the independent or identical assumption, then it must be the normality assumption )? 9.) Look back at the information in the previous question, but assume that you will use 35 samplings to avoid violating the needed assumption to use the central limit theorem. What is the probability of getting a mean evaluation greater than 8? 10.) The law of large numbers says that as the sample size increases the shrinks. (Hint: there may be more than one correct answer) 11.) Suppose we are considering sampling 23 different nursing homes to see how often folks in nursing homes receive visits. The national average is 5 visits per month with a standard deviation of 1.4. a.) What assumptions do we need to hold in order to use the CLT? (Hint: there are 3 of them). b.) Provide an example for a violation of each of the three assumptions in part a. Be sure your examples are relevant to our nursing home context. c.) Find the probability that our sample statistic will be within a 1 visit window of the national average P(4

Answer rating: 100% (QA)

The Answer is in the image, click to view ... View the full answer