Chapter 8 – Normal Probability Distribution

Chapter 8 – Normal Probability Distribution

• A probability distribution in which the random variable is continuous is a continuous probability distribution.

• The normal probability distribution is the most common continuous probability distribution.

Nature of the normal distribution

• Characteristics of normal distribution

1. Bell-shaped with a single peak

2. Symmetrical so two halves are mirror images

•Look at figure 8-3 on page 164•There are numerous normal distributions that

have the same mean, but different standard deviations.•Look at figure 8-4 on page 165.

Importance of the normal distribution.

• The normal distribution is very important for two good reasons.

• 1. It can be used as an approximation for many other distributions.

• 2. Many random variables in the real world follow a normal distribution.

The standard normal distribution• Any normal distribution with a mean and a

standard deviation can be converted to a standard normal distribution. The standard normal distribution has a mean of zero and a standard deviation of one.

• So the standard normal distribution looks like the one shown below.

Standard Distribution (con’t)• Once converted to the standard normal

distribution, the random variable is denoted by Z. The conversion is done by using the following formula:

• Z=(X-)/. Formula 8-1 p. 167

• Where X is the original random variable with a mean of and a standard deviation of .

Standard Distribution (con’t)

• Probability of X being greater than 700 is the same as the probability of Z being greater than 2.

• P(X>700) = P(Z>2)

Standard Distribution (con’t)

• Example Problems 8-1, p. 167

• Z = (700-500) / 100 = 2

300 400 500 600 700 X-Scale

-2 -1 0 1 2 Z-Scale

Areas under the normal curve• Areas under the normal curve can be found

by using appendix D, p. 478.

• Let’s remember a few things1. The area under the normal curve totals 100%.

2. Since the normal curve is symmetrical, 50% of the area is to the right of the mean and the other 50% to the left.

Finding the area• Example problem 8-3, p. 168, • P(Z >1.64)=1- P(Z < 1.64)

=1- 0.9495=0.0505• Example Problem 8-4, page 169• P(Z > -1.65) = P(Z < 1.65) = 0.9505• Example Problem 8-5, p. 169-170• Example Problem 8-6, page 170• Problem #4, page 173• Problem #7, page 174• Problem #10, page 174• Problem #11, page 174

Applications of Z-score• By now , we know how to use appendix D

for finding probabilities. Let’s solve some real-life problems using appendix D.

• Example problem 8-10, p. 175• Example Problem 8-11, p. 176• Problem#6, p. 180• Problem #10, p. 181• Problem #14, p. 181

Sampling Distribution of Mean

• If several samples of size n are taken from a population (whose mean is and standard deviation is ) and their means are computed, these means are normally distributed with a mean of and a standard deviation of /n.

Central Limit Theorem• Calculation of probabilities for sample

_mean, X: _ X - Z = --------

/ n Formula 8-6, Page 189•Example Problem 8-16 (Page 190) Problem #5 (Page 191-192), Problem #11 (Page

192-193)