Transcript Chapter 7 The Normal Probability Distribution

Chapter 7
The Normal Probability
Distribution
7.1
Properties of the Normal Distribution
EXAMPLE
Illustrating the Uniform Distribution
Suppose that United Parcel Service is supposed to
deliver a package to your front door and the arrival time is
somewhere between 10 am and 11 am. Let the random
variable X represent the time from10 am when the
delivery is supposed to take place. The delivery could be
at 10 am (x = 0) or at 11 am (x = 60) with all 1-minute
interval of times between x = 0 and x = 60 equally likely.
That is to say your package is just as likely to arrive
between 10:15 and 10:16 as it is to arrive between 10:40
and 10:41. The random variable X can be any value in
the interval from 0 to 60, that is, 0 < X < 60. Because any
two intervals of equal length between 0 and 60, inclusive,
are equally likely, the random variable X is said to follow a
uniform probability distribution.
Probability Density Function
A probability density function is an equation that
is used to compute probabilities of continuous
random variables that must satisfy the following
two properties. Let f(x) be a probability density
function.
1. The area under the graph of the equation over all
possible values of the random variable must equal
one,that is  f ( x)dx  1
xS
2. The graph of the equation must be greater than or
equal to zero for all possible values of the random
variable. That is, the graph of the equation must lie
on or above the horizontal axis for all possible values
of the random variable, i.e. f(x)>=0
The area under the graph of a density
function over some interval represents
the probability of observing a value of the
random variable in that interval.
EXAMPLE Area as a Probability
Referring to the earlier example, what is the
probability that your package arrives between
10:10 am and 10:20 am?
(10:20-10:10)/60=10/60=1/6
Relative frequency histograms that are symmetric
and bell-shaped are said to have the shape of a
normal curve.
If a continuous random variable is normally
distributed or has a normal probability
distribution, then a relative frequency histogram
of the random variable has the shape of a
normal curve (bell-shaped and symmetric).
Properties of the Normal Density Curve
7. The Empirical Rule: About 68% of the area
under the graph is within one standard deviation
of the mean; about 95% of the area under the
graph is within two standard deviations of the
mean; about 99.7% of the area under the graph
is within three standard deviations of the mean.
EXAMPLE A Normal Random Variable
The following data represent the heights (in
inches) of a random sample of 50 two-year
old males.
(a) Create a relative frequency distribution
with the lower class limit of the first class
equal to 31.5 and a class width of 1.
(b) Draw a histogram of the data.
(c ) Do you think that the variable “height of
2-year old males” is normally distributed?
36.0
34.7
34.4
33.2
35.1
38.3
37.2
36.2
33.4
35.7
36.1
35.2
33.6
39.3
34.8
37.4
37.9
35.2
34.4
39.8
36.0
38.2
39.3
35.6
36.7
37.0
34.6
31.5
34.0
33.0
36.0
37.2
38.4
37.7
36.9
36.8
36.0
34.8
35.4
36.9
35.1
33.5
35.7
35.7
36.8
34.0
37.0
35.0
35.7
38.9
In the next slide, we have a normal
density curve drawn over the histogram.
How does the area of the rectangle
corresponding to a height between 34.5
and 35.5 inches relate to the area under
the curve between these two heights?
EXAMPLE
Interpreting the Area Under a Normal
Curve
The weights of pennies minted after 1982 are
approximately normally distributed with mean 2.46
grams and standard deviation 0.02 grams.
(a) Draw a normal curve with the parameters
labeled.
(b) Shade the region under the normal curve
between 2.44 and 2.49 grams.
(c) Suppose the area under the normal curve for
the shaded region is 0.7745. Provide two
interpretations for this area.
EXAMPLE Relation Between a Normal
Random Variable and a Standard
Normal Random Variable
The weights of pennies minted after 1982 are
approximately normally distributed with mean 2.46
grams and standard deviation 0.02 grams.
Draw a graph that demonstrates the area under the
normal curve between 2.44 and 2.49 grams is
equal to the area under the standard normal curve
between the Z-scores of 2.44 and 2.49 grams.
Chapter 7
The Normal Probability
Distribution
7.2
The Standard Normal Distribution
Properties of the Normal Density Curve
7. The Empirical Rule: About 68% of the area
under the graph is between -1 and 1; about 95%
of the area under the graph is between -2 and 2;
about 99.7% of the area under the graph is
between -3 and 3.
The table gives the area under the standard
normal curve for values to the left of a specified
Z-score, zo, as shown in the figure.
EXAMPLE Finding the Area Under the
Standard Normal Curve
Find the area under the standard normal curve to
the left of Z = -0.38.
Area under the normal curve to the
right of zo
= 1 – Area to the left of zo
EXAMPLE Finding the Area Under the
Standard Normal Curve
Find the area under the standard normal curve to
the right of Z = 1.25.
Look in the Normal distribution table:
P(x>=1.25) =1-P(x<1.25)=1-0.8944=0.1056
EXAMPLE Finding the Area Under the
Standard Normal Curve
Find the area under the standard normal curve
between Z = -1.02 and Z = 2.94.
P(-1.02<x<2.94)=P(x<2.94)-p(x<-1.02)
=0.9984-0.1539
=0.8445
EXAMPLE Finding a Z-score from a Specified
Area to the Left
Find the Z-score such that the area to the left of
the Z-score is 0.68.
i.e., find Z such that P(x<Z)=0.68 , Z =0.46
EXAMPLE Finding a Z-score from a Specified
Area to the Right
Find the Z-score such that the area to the right
of the Z-score is 0.3021.
P(x>Z) =0.3021, so P(x<Z) = 0.6979
z=0.52
EXAMPLE Finding a Z-score
EXAMPLE Finding the Value of z
Find the value of z0.25
Notation for the Probability of a Standard Normal
Random Variable
P(a < Z < b)
represents the probability a
standard normal random variable is
between a and b
P(Z > a)
represents the probability a
standard normal random variable is
greater
P(Z < a)
than a.
represents the probability a standard
normal random variable is less than
a.
EXAMPLE Finding Probabilities of Standard
Normal Random Variables
Find each of the following probabilities:
(a) P(Z < -0.23)
(b) P(Z > 1.93)
(c) P(0.65 < Z < 2.10)