288 Chapter 7 The Normal Distribution Check Your Understanding 1. Following is a probability density curve with the area between 0 and 1 and the area between 1 and 2 indicated. Area = 0.63 0 1 2 3 4 5 1.0 0.8 0.6 0.4 0.2 0 Area = 0.23 a. What proportion of the population is between 0 and 1? 0.63 b. What is the probability that a randomly selected value will be between 1 and 2? 0.23 c. What proportion of the population is between 0 and 2? 0.86 d. What is the probability that a randomly selected value will be greater than 2? 0.14 Answers are on page 299. Objective 2 Use a normal curve to describe a normal population The Normal Distribution Probability density curves come in many varieties, depending on the characteristics of the populations they represent. Remarkably, many important statistical procedures can be carried out using only one type of probability density curve, called a normal curve. A population that is represented by a normal curve is said to be normally distributed, or to have a normal distribution. Figure 7.4 presents some examples of normal curves. Explain It Again The mode of a curve: Recall that a peak in a histogram is called a mode of the histogram. Similarly, a peak in a probability density curve, such as a normal curve, is called a mode of the curve. The location and shape of a normal curve reflect the mean and standard deviation of the population. The curve is symmetric around its peak, or mode. Therefore, the mode is equal to the population mean. The population standard deviation measures the spread of the population. Therefore, the normal curve is wide and flat when the population standard deviation is large, and tall and narrow when the population standard deviation is small. 0.5 0.4 0.3 0.2 0.1 0 μ = 0, σ = 1 μ = 0, σ = 2 −10 −5 0 5 10 0.5 0.4 0.3 0.2 0.1 0 μ = 0, σ = 1 μ = 5, σ = 1 −5 0 5 10 (a) (b) Figure 7.4 (a) Both populations have mean 0. The population with standard deviation 2 is more spread out than the population with standard deviation 1. (b) Both populations have the same spread, because they have the same standard deviation. The curves are centered over the population means.
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