Transcript 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
-2
400
-1
500
0
600
1
700
2
X-Scale
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 things
1. 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)