234 CHAPTER 5 Normal Probability Distributions Introduction to Normal Distributions and the Standard Normal Distribution 5.1 What You Should Learn How to interpret graphs of normal probability distributions How to find areas under the standard normal curve Properties of a Normal Distribution The Standard Normal Distribution Properties of a Normal Distribution In Section 4.1, you distinguished between discrete and continuous random variables, and learned that a continuous random variable has an infinite number of possible values that can be represented by an interval on a number line. Its probability distribution is called a continuous probability distribution. In this chapter, you will study the most important continuous probability distribution in statistics—the normal distribution. Normal distributions can be used to model many sets of measurements in nature, industry, and business. For instance, the systolic blood pressures of humans, the lifetimes of smartphones, and housing costs are all normally distributed random variables. A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. A normal distribution has these properties. 1. The mean, median, and mode are equal. 2. The normal curve is bell-shaped and is symmetric about the mean. 3. The total area under the normal curve is equal to 1. 4. The normal curve approaches, but never touches, the x@axis as it extends farther and farther away from the mean. 5. Between m - s and m + s (in the center of the curve), the graph curves downward. The graph curves upward to the left of m - s and to the right of m + s. The points at which the curve changes from curving upward to curving downward are called inflection points. − μ σ +μ σ μ Inflection points Total area = 1 − μ σ2 − μ σ3 +μ σ2 +μ σ3 x DEFINITION You have learned that a discrete probability distribution can be graphed with a histogram. For a continuous probability distribution, you can use a probability density function (pdf). A probability density function has two requirements: (1) the total area under the curve is equal to 1, and (2) the function can never be negative. Study Tip A normal curve with mean m and standard deviation s can be graphed using the normal probability density function y = 1 s22p e-1x-m2 2 (2s 2). (This formula will not be used in the text.) Because e ≈ 2.718 and p ≈ 3.14, a normal curve depends completely on m and s. For help with division involving square roots, see Integrated Review at MyLab® Statistics
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