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ch06 - blu03683_ch06.qxd 01:51 PM Page 285 CHAPTE R 6 The...

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6–1 Objectives After completing this chapter, you should be able to 1 Identify distributions as symmetric or skewed. 2 Identify the properties of a normal distribution. 3 Find the area under the standard normal distribution, given various z values. 4 Find probabilities for a normally distributed variable by transforming it into a standard normal variable. 5 Find specific data values for given percentages, using the standard normal distribution. 6 Use the central limit theorem to solve problems involving sample means for large samples. 7 Use the normal approximation to compute probabilities for a binomial variable. Outline 6–1 Introduction 6–2 Properties of a Normal Distribution 6–3 The Standard Normal Distribution 6–4 Applications of the Normal Distribution 6–5 The Central Limit Theorem 6–6 The Normal Approximation to the Binomial Distribution 6–7 Summary 6 6 The Normal Distribution C H A P T E R
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286 Chapter 6 The Normal Distribution 6–2 Statistics Today What Is Normal? Medical researchers have determined so-called normal intervals for a person’s blood pressure, cholesterol, triglycerides, and the like. For example, the normal range of sys- tolic blood pressure is 110 to 140. The normal interval for a person’s triglycerides is from 30 to 200 milligrams per deciliter (mg/dl). By measuring these variables, a physician can determine if a patient’s vital statistics are within the normal interval or if some type of treatment is needed to correct a condition and avoid future illnesses. The question then is, How does one determine the so-called normal intervals? See Statistics Today—Revisited at the end of the chapter. In this chapter, you will learn how researchers determine normal intervals for specific medical tests by using a normal distribution. You will see how the same methods are used to determine the lifetimes of batteries, the strength of ropes, and many other traits. 6–1 Introduction Random variables can be either discrete or continuous. Discrete variables and their dis- tributions were explained in Chapter 5. Recall that a discrete variable cannot assume all values between any two given values of the variables. On the other hand, a continuous variable can assume all values between any two given values of the variables. Examples of continuous variables are the heights of adult men, body temperatures of rats, and cho- lesterol levels of adults. Many continuous variables, such as the examples just mentioned, have distributions that are bell-shaped, and these are called approximately normally dis- tributed variables. For example, if a researcher selects a random sample of 100 adult women, measures their heights, and constructs a histogram, the researcher gets a graph similar to the one shown in Figure 6–1(a). Now, if the researcher increases the sample size and decreases the width of the classes, the histograms will look like the ones shown in Figure 6–1(b) and (c). Finally, if it were possible to measure exactly the heights of all
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