Transcript The Normal Distribution

Continuous Random
Variables
Chapter 6
Overview
This chapter will deal with the
construction of discrete probability
distributions by combining methods of
descriptive statistics from Chapters 2 and
3 and those of probability presented in
Chapter 4.
 A probability distribution, in general, will
describe what will probably happen
instead of what actually did happen

Combining Descriptive Methods
and Probabilities
In this chapter we will construct probability distributions by presenting
possible outcomes along with the relative frequencies we expect.
Why do we need probability
distributions?

Many decisions in business, insurance,
and other real-life situations are made by
assigning probabilities to all possible
outcomes pertaining to the situation and
then evaluating the results
◦ Saleswoman can compute probability that she will make
0, 1, 2, or 3 or more sales in a single day. Then, she
would be able to compute the average number of sales
she makes per week, and if she is working on
commission, she will be able to approximate her weekly
income over a period of time.
Discrete Variables (Data)—
Chapter 5
Can be assigned
values such as 0, 1, 2,
3
 “Countable”
 Examples:

 Number of children
 Number of credit cards
 Number of calls received
by switchboard
 Number of students
Remember
Continuous Variables (Data)--Chapter 6
OUR FOCUS
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Can assume an infinite
number of values
between any two specific
values
Obtained by measuring
Often include fractions
and decimals
Examples:
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Temperature
Height
Weight
Time
6.1 Introduction to the Normal Curve
 6.2 Reading a Normal Curve Table
 6.3 Finding the Probability using the
Normal Curve
 6.4 Find z-values using the Normal Curve
 6.5 Find t-Values using the Student tdistribution
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Outline

Objectives:
◦ Identify the properties of a normal distribution
Section 6.1 Introduction to the
Normal Curve

A normal distribution is a continuous,
symmetric, bell-shaped distribution of a
2

(
x


)
variable
f ( x) 
e
2 2
 2
where e  2.718
  3.14
What is a Normal Distribution?

Any particular normal distribution is
determined by two parameters
◦ Mean, 
◦ Standard Deviation, 

A normal distribution is bell-shaped and is
symmetric
 Symmetry of the curve means that if you cut
the curve in half, the left and right sides are
mirror images (the line of symmetry is x = )
 Bell shaped means that the majority of the
data is in the middle of the distribution and the
amount tapers off evenly in both directions
from the center
 There is only one mode (unimodal)
 Mean = Median = Mode
Properties of the Theoretical
Normal Distribution

The total area under
a normal distribution
is equal to 1 or
100%. This fact may
seem unusual, since
the curve never
touches the x-axis,
but one can prove it
mathematically by
using calculus

The area under the
part of the normal
curve that lies within
1 standard deviation
of the mean is
approximately 0.68
or 68%, within 2
standard deviations,
about 0.95 or 95%,
and within 3 standard
deviations, about
0.997 or 99.7%.
(Empirical Rule)
Properties of the Theoretical
Normal Distribution
Introduction

A continuous random variable has a uniform
distribution if its values are spread evenly
over the range of possibilities.
◦ The graph of a uniform distribution results in a
rectangular shape.
◦ A uniform distribution makes it easier to see two
very important properties of a normal distribution
 The area under the graph of a probability distribution is
equal to 1.
 There is a correspondence between area and
probability (relative frequency)
Uniform Distribution*
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Experiment: Roll a die
◦ Create a probability distribution in table form
◦ Sketch graph
◦ Using the graph, find the following
probabilities:
◦
◦
◦
◦
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P(5)
P(a number
P(a number
P(a number
P(a number
less than 4)
between 2 and 6, inclusive)
greater than 3)
less than and including 6)
Example: Roll a die
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A researcher selects a random sample of
100 adult women, measures their heights,
and constructs a histogram.
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Because the total area under the normal
distribution is 1, there is a correspondence
between area and probability
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Since each normal distribution is determined
by its own mean and standard deviation, we
would have to have a table of areas for each
possibility!!!! To simplify this situation, we
use a common standard that requires only
one table.

The standard normal distribution is a
normal distribution with a mean of 0 and
a standard deviation of 1.
Standard Normal Distribution
Draw a picture ALWAYS!!!!!!!
 Shade the area desired.
 Follow given directions to find area (aka
probability) using the calculator
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Area is always a positive number, even if
the z-value is negative (this simply
implies the z-value is below the mean)
Finding Areas Under the Standard
Normal Distribution Curve

Find area under the standard normal
distribution curve
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Between 0 and 1.66
Between 0 and -0.35
To the right of z = 1.10
To the left of z = -0.48
Between z =1.23 and z =1.90
Between z =-0.96 and z =-0.36
To left of z =1.31
To the left of z =-2.15 and to the right of z
=1.62
Examples
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Worksheet
Assignment