Normal Distribution

Topics: Normal distribution, Standard deviation, Probability theory Pages: 129 (16101 words) Published: February 17, 2014
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C H A P T E

R

6

The Normal
Distribution

Objectives

Outline

After completing this chapter, you should be able to

1
2
3

Identify distributions as symmetric or skewed.

4

Find probabilities for a normally distributed
variable by transforming it into a standard
normal variable.

Introduction
6–1

Normal Distributions

Identify the properties of a normal distribution.
Find the area under the standard normal
distribution, given various z values.

5

Find specific data values for given
percentages, using the standard normal
distribution.

6

6–3 The Central Limit Theorem
6–4 The Normal Approximation to the Binomial
Distribution

Use the central limit theorem to solve
problems involving sample means for large
samples.

7

6–2 Applications of the Normal Distribution

Use the normal approximation to compute
probabilities for a binomial variable.

Summary

6–1

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Chapter 6 The Normal Distribution

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 systolic 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.

Introduction
Random variables can be either discrete or continuous. Discrete variables and their distributions 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 cholesterol levels of adults. Many continuous variables, such as the examples just mentioned, have distributions that are bell-shaped, and these are called approximately normally distributed 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 adult females in the United States and plot them, the histogram would approach what is called a normal distribution, shown in Figure 6–1(d). This distribution is also known as 6–2

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Chapter 6 The Normal Distribution

301

Figure 6–1
Histograms for the
Distribution of Heights
of Adult Women
(a) Random sample of 100 women

(b) Sample size increased and class width decreased

(c) Sample size increased and class width
decreased further

(d) Normal distribution for the population

Figure 6–2
Normal and Skewed
Distributions

Mean
Median
Mode
(a) Normal

Mean Median Mode
(b) Negatively skewed

Objective

1

Identify distributions
as symmetric or
skewed.

Mode Median Mean
(c) Positively skewed

a bell curve or a Gaussian...
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