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Larger Sample, More Accurate Estimate. Suppose that, in fact, the total cholesterol level of all men aged 20-34 follows the Normal distribution with mean \(\mu=182\) milligrams per deciliter (mg/dL) and standard deviation \(\sigma=37 \mathrm{mg} / \mathrm{dL}\). a. Choose an SRS of 100 men from this population. What is the sampling distribution of \(x\) ? What is the probability that \(x\) takes a value between 180 and 184 \(\mathrm{mg} / \mathrm{dL}\) ? This is the probability that \(x\) estimates \(\mu\) within \(\pm 2 \mathrm{mg} / \mathrm{dL}\). b. Choose an SRS of 1000 men from this population. Now what is the probability that \(x\) falls within \(\pm 2 \mathrm{mg} / \mathrm{dL}\) of \(\mu\) ? The larger sample is much more likely to give an accurate estimate of \(\mu\).

Annual returns on stocks vary a lot. The long-term mean return on stocks in the S\&P 500 is \(9.8 \%\), and the long-term standard deviation of returns is \(16.8 \%\). The law of large numbers says that a. you can get an average return higher than the mean \(9.8 \%\) by investing in a large number of the \(\mathrm{S} \& \mathrm{P}\) stocks. b. as you invest in more and more stocks chosen at random, your long-term average return on these stocks gets ever closer to \(9.8 \%\). c. if you invest in a large number of stocks chosen at random, your long-term average return will have approximately a Normal distribution.

Roulette. A roulette wheel has 38 slots, of which 18 are black, 18 are red, and two are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. One of the simplest wagers is to choose red or black. A bet of \(\$ 1\) on red returns \(\$ 2\) if the ball lands in a red slot. Otherwise, the player loses the dollar. When gamblers bet on red or black, the two green slots result in losses. Because the probability of winning \(\$ 2\) is \(18 / 38\), the mean payoff from a \(\$ 1\) bet is twice \(18 / 38\), or \(94.7\) cents. Explain what the law of large numbers tells us about what will happen if a gambler makes very many bets on red.

Pollutants in Aut o Ex hausts (continued). The level of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) in the exhaust over the useful life ( 150,000 miles of driving) of cars of a particular model varies Normally with mean 80 \(\mathrm{mg} / \mathrm{mi}\) and standard deviation \(4 \mathrm{mg} / \mathrm{mi}\). A company has 25 cars of this model in its fleet. What is the level \(L\) such that the probability that the average NOX + NMOG level \(x\) for the fleet is greater than \(L\) is only \(0.01\) ? (Hint: This requires a backward Normal calculation. See page 89 in Chapter 3 if you need a review.)