Chapter 11 Data Descriptions and Probability Distributions Section 5 Normal Distribution

Chapter 11

Data Descriptions and Probability Distributions

Section 5

Normal Distribution

2Barnett/Ziegler/Byleen Finite Mathematics 12e

Learning Objectives for Section 11.5 Normal Distributions

The student will be able to identify what is meant by a normal distribution.

The student will be able to find the area under normal curves.

The student will be able to approximate the binomial distribution with a normal distribution.


Normal Distributions

We have seen that the histogram for a binomial distribution with n = 20 trials and p = 0.50 was shaped like a bell if we join the tops of the rectangles with a smooth curve.

Real world data, such as weights of individuals, heights, test scores have histograms that have a symmetric bell shape. We call such distributions normal distributions. They will be the focus of this section.


Mathematicians and the Normal Curve

Three mathematicians contributed to the mathematical foundation for this curve. They are Abraham De Moivre, Pierre Laplace and Carl Friedrich Gauss.


Abraham De Moivre

De Moivre pioneered the development of analytic geometry and the theory of probability. He published The Doctrine of Chance in 1718. The definition of statistical independence appears in this book together with many problems with dice and other games. He also investigated mortality statistics and the foundation of the theory of annuities De Moivre

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/De_Moivre.html


Pierre Laplace

Laplace systematized and elaborated probability theory in "Essai Philosophique sur les Probabilités" (Philosophical Essay on Probability, 1814). He was the first to publish the value of the Gaussian integral.

We will talk about Gauss later.

Laplace


Bell-Shaped Curves

Many frequency distributions have a symmetric, bell shaped histogram.

Example 1: The frequency distribution of heights of males is symmetric about a mean of 69.5 inches.

Example 2: IQ scores are symmetrically distributed about a mean of 100, with a standard deviation of 15 or 16. The frequency distribution of IQ scores is bell shaped.

Example 3: SAT test scores have a bell shaped, symmetric distribution.


Properties of Normal Curves

Normal curves are bell-shaped and are symmetrical with respect to the vertical line x = (the mean).

The curve approaches, but does not touch, the horizontal axis as x gets very large (or x gets very small)

The shape of a normal curve is completely determined by its mean and standard deviation - a small standard deviation indicates a tight clustering about the mean and thus a tall, narrow curve; a large standard deviation indicates a large deviation from the mean and thus a broad, flat curve.


Graphs of Normal Curves

Several normal curves

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-10 -5 0 5 10


Probability and Area under the Normal Curve

Key fact: For a normally distributed variable, the percentage of observations that lie within a specified range equals the corresponding area under its associated normal curve.

This is approximately true for a variable that is approximately normally distributed.

P(a < x < b) = probability that the random variable X is between a and b = area under the normal curve between x = a and x = b.

The total area under a normal curve is 1.


Finding Areas Under a Normal Curve

Finding the area under a normal curve between x = a and x = b requires calculus. We can circumvent this problem by looking up the values in a table.

However, the shape of each normal curve is determined by the standard deviation; the greater the standard deviation, the “flatter” and more spread out the normal curve will be. We would need infinitely many tables.

The solution is to standardize a normally distributed variable, and to use the table for the standard normal curve.


Standard Normal Distribution

The standard normal distribution has a mean of 0 and a standard deviation of 1. Values on the horizontal axis are called z values. Values on the y axis are probabilities and are decimal numbers between 0 and 1, inclusive.


Areas under the Standard Normal Curve

For the following examples,

1. Draw a diagram

2. Shade the appropriate area

3. Use a table or a TI 83 to find the probability

(A) Find P(0 < z < 1.2)


Example 2

A traffic study at one point on an interstate highway shows that vehicle speeds are normally distributed with a mean of 61.3 mph and a standard deviation of 3.3 miles per hour. If a vehicle is randomly checked, find the probability that its speed is between 55 and 60 miles per hour.


Step 2. Convert x score to a standardized z score:

Step 3. Use table or TI 83 to find area. Answer: 0.4719 - 0.1517 = 0.3202

Example 2(continued)

Step 1. Draw a normal curve and shade appropriate area.

State the probability:

P(55 < x < 60)

where x is speed.

55 61.3 60 61.3

3.3 3.3P z

= P(–1.91 < z < -0.39)


Mathematical Equation for Bell-Shaped Curves

Carl Friedrich Gauss, a mathematician, was probably the first to realize that certain data had bell-shaped distributions. He determined that the following equation could be used to describe these distributions:

where , are the mean and standard deviation of the data.

2

2( )21( )

2

x

f x e


Using the Normal Curve to Approximate Binomial Probabilities

Example: Find the probability of getting 12 or more Heads when you toss a coin 20 times.



Example: Find the probability of getting 12 or more Heads when you toss a coin 20 times.Solution: We have seen that the histogram for a binomial distribution with n = 20 trials and p = 0.50 was shaped like a bell if we join the tops of the rectangles with a smooth curve.To find the probability that x (number of heads) is greater than or equal to 12, we would have to calculate P(x=12) + P(x=13) + P(x=14) + …+ P(x=20) . The calculations would be very tedious, to say the least.



We could, instead, treat the binomial distribution as a normal curve, since its shape is pretty close to being a bell-shaped curve, and then find the probability that x is greater than 12 using the procedure for finding areas under a normal curve.

= np = 10 = sqrt(np(1–p)) = sqrt(5) = 2.24

Probability that x ≥ 12 = total area in yellow


Approximating Binomial by Normal(continued)

Because the normal curve is continuous and the binomial distribution is discrete, we have to make what is called a correction for continuity.

Since we want P(x ≥ 12), we must include the rectangular area corresponding to x = 12. The base of this rectangle starts at 11.5 and ends at 12.5.

Therefore, we must find P(x > 11.5). The rectangle representing the P(x = 12) extends from 11.5 to 12.5 on the horizontal axis.


Approximating Binomial by Normal(continued)

Using the procedure for finding area under a non-standard normal curve we get

= P(z > 0.770) = 0.5 – 0.2794 = 0.2206

11.5 10( 11.5)

2.24P x P z

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Chapter 11 Data Descriptions and Probability Distributions Section 5 Normal Distribution