286 Chapter 7 The Normal Distribution SECTION 7.1 The Standard Normal Curve Objectives 1. Use a probability density curve to describe a population 2. Use a normal curve to describe a normal population 3. Find areas under the standard normal curve 4. Find z-scores corresponding to areas under the normal curve Objective 1 Use a probability density curve to describe a population Figure 7.1, first shown in Section 2.2, presents a relative frequency histogram for the emissions of a sample of 65 vehicles. The amount of emissions is a continuous variable, because its possible values are not limited to some discrete set. The class intervals are chosen so that each rectangle represents a reasonably large number of vehicles. If the sample were larger, we could make the rectangles narrower. In particular, if we had information on the entire population, containing millions of vehicles, we could make the rectangles extremely narrow. The histogram would then look quite smooth and could be approximated by a curve, which might look like Figure 7.2. 0 1 2 3 4 5 6 7 0.4 0.3 0.2 0.1 0 Relative Frequency Particulate Emissions Figure 7.1 Relative frequency histogram for the emissions of a sample of 65 vehicles 0 1 2 3 4 5 6 7 0.4 0.3 0.2 0.1 0 Particulate Concentration Figure 7.2 The histogram for a large population of vehicles could be drawn with extremely narrow rectangles, and could be represented by a curve. If a vehicle were chosen at random from this population to have its emissions measured, the emissions level would be a continuous random variable. The curve used to describe the distribution of a continuous random variable is called the probability density curve of the random variable. The probability density curve tells us what proportion of the population falls within any given interval. For example, Figure 7.3 illustrates the proportion of the population of vehicles whose emissions levels are between 3 and 4. In general, the area under a probability density curve between any two values a and b has two interpretations: It represents the proportion of the population whose values are between a and b, and it also represents the probability that a randomly selected value from the population will be between a and b. The area of the shaded region is equal to the proportion of the population with values between 3 and 4. 0 1 2 3 4 5 6 7 0.4 0.3 0.2 0.1 0 Particulate Concentration Figure 7.3 The area under a probability density curve between two values is equal to the proportion of the population that falls between the two values.
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