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AP Statistics Central Limit Theorem Worksheet 1. A bottling company uses a filling machine to fill plastic bottles with a popular cola. The bottles are supposed to contain 300 millilters (ml). In fact, the contents vary according to a normal distribution with mean = 303 ml and standard deviation = 3 ml. a. What is the probability that an individual bottle contains less than 300 ml? b. Now take a random sample of 10 bottles. What are the mean and standard deviation of the sample mean contents x-bar of these 10 bottles? c. What is the probability that the sample mean contents of the 10 bottles is less than 300 ml? 2. For 1998 as a whole, the mean return of all common stocks listed on the New York Stock Exchange (NYSE) was = 16% and standard deviation = 26%. Assume that the distribution of returns is roughly normal. a. What % of stocks lost money? b. Suppose we create a portfolio of 8 stocks by randomly selecting stocks from the NYSE and investing equal amounts of money in each stock. What are the mean and standard deviation of the sample mean returns xbar for these 8 stocks? c. What is the probability the portfolio loses money? Explain the difference between this result and that of part (a). d. The probability is 0.05 that a portfolio constructed this way has a return of more than ________ ? (This would be the 95th percentile of portfolio returns; however, remember that these portfolios form a hypothetical population -- no one actually owns such a portfolio.) 3. The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 264 days and standard deviation 16 days. (See a previous worksheet for some questions involving this distribution.) Consider 15 pregnant women from a rural area. Assume they are equivalent to a random sample from all women. a. What are the mean and standard deviation of the sample mean length of pregnancy x-bar of these 15 pregnancies? b. If we want to predict, with 90% accuracy, the sample mean length of pregnancy for 15 randomly selected women, what values do we use? (That is, find value L AND U such that there's a 90% probability the sample mean x-bar lies between L and U.) c. What's the probability the sample mean length of pregnancy lasts less than 250 days? (Contrast this with the probability a single pregnant women is pregnant for less than 250 days, which is 0.1908.) d. Toxic waste is believed to have effected the health of residents of this area. Suppose the sample mean length of pregnancy is indeed 250 days; use the result of part (c) to argue that the waste has an effect of length of pregnancy.