248 CHAPTER 6 Normal Probability Distributions 0 0.2 1 2 Area 5 1 3 4 5 x (waiting time in minutes) P(x) FIGURE 6-2 Uniform Distribution of Waiting Time YOUR TURN. Do Exercise 5 “Continuous Uniform Distribution.” Waiting Times for Airport Security EXAMPLE 2 Given the uniform distribution illustrated in Figure 6-2, find the probability that a randomly selected passenger has a waiting time of at least 2 minutes. YOUR TURN. Do Exercise 7 “Continuous Uniform Distribution.” 0 0.2 1 2 Area 5 0.2 3 3 5 0.6 3 4 5 x (waiting time in minutes) P(x) FIGURE 6-3 Using Area to Find Probability SOLUTION The shaded area in Figure 6-3 represents waiting times of at least 2 minutes. Because the total area under the density curve is equal to 1, there is a correspondence between area and probability. We can easily find the desired probability by using areas as follows: P1wait time of at least 2 min2 = height * width of shaded area in Figure 6-3 = 0.2 * 3 = 0.6 INTERPRETATION The probability of randomly selecting a passenger with a waiting time of at least 2 minutes is 0.6. Standard Normal Distribution The density curve of a uniform distribution is a horizontal straight line, so we can find the area of any rectangular region by applying this formula: Area = height * width. Because the density curve of a normal distribution has a more complicated bell shape, as shown in Figure 6-1, it is more difficult to find areas. However, the basic principle Power of Small Samples The Environmental Protection Agency (EPA) had discovered that Chrysler automobiles had malfunctioning carburetors, with the result that carbon monoxide emissions were too high. Chryslers with 360- and 400-cubic-inch displacements and two-barrel carburetors were involved. The EPA ordered Chrysler to fix the problem, but Chrysler refused, and the case of Chrysler Corporation vs. The Environmental Protection Agency followed. That case led to the conclusion that there was “substantial evidence” that the Chryslers produced excessive levels of carbon monoxide. The EPA won the case, and Chrysler was forced to recall and repair 208,000 vehicles. In discussing this case in an article in AMSTAT News, Chief Statistician for the EPA Barry Nussbaum wrote this: “Sampling is expensive, and environmental sampling is usually quite expensive. At the EPA, we have to do the best we can with small samples or develop models. . . . What was the sample size required to affect such a recall (of the 208,000 Chryslers)? The answer is a mere 10. It is both an affirmation of the power of inferential statistics and a challenge to explain how such a (small) sample could possibly suffice.” T t ( d t a
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