Download MATH 105: Finite Mathematics 9

Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
MATH 105: Finite Mathematics
9-6: The Normal Distribution
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Outline
1
Bernoulli Probability and the Normal Distribution
2
Properties of the Normal Distribution
3
Examples
4
Conclusion
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Outline
1
Bernoulli Probability and the Normal Distribution
2
Properties of the Normal Distribution
3
Examples
4
Conclusion
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Graphing Bernoulli Probability Distributions
We start this section by examining the “Probability Distribution”
we get from a Bernoulli Process. This is called the Binomial
Probability Distribution.
Example
Construct a probability histogram for a Bernoulli process with
n = 6 trials where p = 12
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Graphing Bernoulli Probability Distributions
We start this section by examining the “Probability Distribution”
we get from a Bernoulli Process. This is called the Binomial
Probability Distribution.
Example
Construct a probability histogram for a Bernoulli process with
n = 6 trials where p = 12
r
0
C (6, 0)
1
C (6, 1)
2
3
4
5
6
Pr[r ]
“ ”0 “ ”6
1
1
“ 2 ”1 “ 2 ”5
1
“ 2 ”2
C (6, 2) 12
“ ”3
C (6, 3) 12
“ ”4
C (6, 4) 12
“ ”5
C (6, 5) 12
“ ”6
C (6, 6) 12
1
“ 2 ”4
1
“ 2 ”3
1
“ 2 ”2
1
“ 2 ”1
1
“ 2 ”0
1
2
≈ 0.01563
≈ 0.09375
≈ 0.23438
≈ 0.31250
≈ 0.23438
≈ 0.09375
≈ 0.01563
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Graphing Bernoulli Probability Distributions
We start this section by examining the “Probability Distribution”
we get from a Bernoulli Process. This is called the Binomial
Probability Distribution.
Example
Construct a probability histogram for a Bernoulli process with
n = 6 trials where p = 12
r
Pr[r ]
“ ”0 “ ”6
1
0
C (6, 0)
1
C (6, 1) 12
“ ”2
C (6, 2) 12
“ ”3
C (6, 3) 12
“ ”4
C (6, 4) 12
“ ”5
C (6, 5) 12
“ ”6
C (6, 6) 12
2
3
4
5
6
1
“ 2 ”1 “ 2 ”5
1
“ 2 ”4
1
“ 2 ”3
1
“ 2 ”2
1
“ 2 ”1
1
“ 2 ”0
1
2
≈ 0.01563
≈ 0.09375
≈ 0.23438
≈ 0.31250
≈ 0.23438
≈ 0.09375
≈ 0.01563
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
As n Increases. . .
As n increases, one could almost say the number of successes
becomes more like a continuous variable. The picture for n = 100
is shown below.
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
As n Increases. . .
As n increases, one could almost say the number of successes
becomes more like a continuous variable. The picture for n = 100
is shown below.
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
The Normal Distribution
As n continues to increase, the Binomial Distribution approaches
the Normal Distribution which is shown below.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
The Normal Distribution
As n continues to increase, the Binomial Distribution approaches
the Normal Distribution which is shown below.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Outline
1
Bernoulli Probability and the Normal Distribution
2
Properties of the Normal Distribution
3
Examples
4
Conclusion
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
General Properties
Properties of the Normal Distribution
The following are properties of the Normal Distribution.
Bell shaped
Symmetric about µ
Probability = Area
Area Under Curve = 1
Probability a value lies between a and b is area under curve
between a and b.
Standard normal distribution has µ = 0 and σ = 1.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
General Properties
Properties of the Normal Distribution
The following are properties of the Normal Distribution.
Bell shaped
Symmetric about µ
Probability = Area
Area Under Curve = 1
Probability a value lies between a and b is area under curve
between a and b.
Standard normal distribution has µ = 0 and σ = 1.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
General Properties
Properties of the Normal Distribution
The following are properties of the Normal Distribution.
Bell shaped
Symmetric about µ
Probability = Area
Area Under Curve = 1
Probability a value lies between a and b is area under curve
between a and b.
Standard normal distribution has µ = 0 and σ = 1.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
General Properties
Properties of the Normal Distribution
The following are properties of the Normal Distribution.
Bell shaped
Symmetric about µ
Probability = Area
Area Under Curve = 1
Probability a value lies between a and b is area under curve
between a and b.
Standard normal distribution has µ = 0 and σ = 1.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
General Properties
Properties of the Normal Distribution
The following are properties of the Normal Distribution.
Bell shaped
Symmetric about µ
Probability = Area
Area Under Curve = 1
Probability a value lies between a and b is area under curve
between a and b.
Standard normal distribution has µ = 0 and σ = 1.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
General Properties
Properties of the Normal Distribution
The following are properties of the Normal Distribution.
Bell shaped
Symmetric about µ
Probability = Area
Area Under Curve = 1
Probability a value lies between a and b is area under curve
between a and b.
Standard normal distribution has µ = 0 and σ = 1.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
General Properties
Properties of the Normal Distribution
The following are properties of the Normal Distribution.
Bell shaped
Symmetric about µ
Probability = Area
Area Under Curve = 1
Probability a value lies between a and b is area under curve
between a and b.
Standard normal distribution has µ = 0 and σ = 1.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Empirical Rule
Empirical Rule
In a normal distribution, approximately
1
68% of outcomes within 1 standard deviation of the mean.
2
95% of outcomes within 2 standard deviations of the mean.
3
99.7% of outcomes within 3 standard deviations of the mean.
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Empirical Rule
Empirical Rule
In a normal distribution, approximately
1
68% of outcomes within 1 standard deviation of the mean.
2
95% of outcomes within 2 standard deviations of the mean.
3
99.7% of outcomes within 3 standard deviations of the mean.
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Applying the Empirical Rule
Example
A standardized test has a mean score of µ = 125 and a standard
deviation of σ = 12. 1200 students take the test.
1
How many scored between 113 and 137?
2
How many scored above 125?
3
How many students scored less than 89?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Applying the Empirical Rule
Example
A standardized test has a mean score of µ = 125 and a standard
deviation of σ = 12. 1200 students take the test.
1
How many scored between 113 and 137?
2
How many scored above 125?
3
How many students scored less than 89?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Applying the Empirical Rule
Example
A standardized test has a mean score of µ = 125 and a standard
deviation of σ = 12. 1200 students take the test.
1
How many scored between 113 and 137?
0.68 × 1200 = 816
2
How many scored above 125?
3
How many students scored less than 89?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Applying the Empirical Rule
Example
A standardized test has a mean score of µ = 125 and a standard
deviation of σ = 12. 1200 students take the test.
1
How many scored between 113 and 137? (816)
2
How many scored above 125?
3
How many students scored less than 89?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Applying the Empirical Rule
Example
A standardized test has a mean score of µ = 125 and a standard
deviation of σ = 12. 1200 students take the test.
1
How many scored between 113 and 137? (816)
2
How many scored above 125?
0.5 × 1200 = 600
3
How many students scored less than 89?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Applying the Empirical Rule
Example
A standardized test has a mean score of µ = 125 and a standard
deviation of σ = 12. 1200 students take the test.
1
How many scored between 113 and 137? (816)
2
How many scored above 125? (600)
3
How many students scored less than 89?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Applying the Empirical Rule
Example
A standardized test has a mean score of µ = 125 and a standard
deviation of σ = 12. 1200 students take the test.
1
How many scored between 113 and 137? (816)
2
How many scored above 125? (600)
3
How many students scored less than 89?
1
× (1 − .997) × 1200 ≈ 2
2
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Other Shapes
As the mean and standard deviation of a normal distribution
changes, so does the shape of the graph.
µ shifts center
σ changes spread
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Other Shapes
As the mean and standard deviation of a normal distribution
changes, so does the shape of the graph.
µ shifts center
σ changes spread
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Other Shapes
As the mean and standard deviation of a normal distribution
changes, so does the shape of the graph.
µ shifts center
σ changes spread
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Other Shapes
As the mean and standard deviation of a normal distribution
changes, so does the shape of the graph.
µ shifts center
σ changes spread
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Using the z-score
Since there are infinitely many normal distributions, we will convert
each to the standard normal distribution with µ = 0 and σ = 1.
z-scores
If x is a data point in a normal distribution with mean µ and
standard deviation σ, then the z-score for x is:
z=
x −µ
σ
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Using the z-score
Since there are infinitely many normal distributions, we will convert
each to the standard normal distribution with µ = 0 and σ = 1.
z-scores
If x is a data point in a normal distribution with mean µ and
standard deviation σ, then the z-score for x is:
z=
x −µ
σ
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Using the z-score
Since there are infinitely many normal distributions, we will convert
each to the standard normal distribution with µ = 0 and σ = 1.
z-scores
If x is a data point in a normal distribution with mean µ and
standard deviation σ, then the z-score for x is:
z=
x −µ
σ
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Using the z-score
Since there are infinitely many normal distributions, we will convert
each to the standard normal distribution with µ = 0 and σ = 1.
z-scores
If x is a data point in a normal distribution with mean µ and
standard deviation σ, then the z-score for x is:
z=
x −µ
σ
Use the table in the back of your
book to determine the area under the curve between the mean
and a given z-score.
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Outline
1
Bernoulli Probability and the Normal Distribution
2
Properties of the Normal Distribution
3
Examples
4
Conclusion
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(a) How many standard deviations away from the mean are trees
which are 300 and 120 inches tall?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(a) How many standard deviations away from the mean are trees
which are 300 and 120 inches tall?
z=
z=
300 − 240
= 1.50
40
120 − 240
= −3.00
40
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(b) What percent of the trees are between 240 and 300 inches tall?
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(b) What percent of the trees are between 240 and 300 inches tall?
µ = 240
300 − 240
= 1.50
40
A = 0.4332 ⇒ 43.3%
z=
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(c) What percent of the trees are greater than 300 inches tall?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(c) What percent of the trees are greater than 300 inches tall?
z=
300 − 240
= 1.50
40
A = 1−0.4332 = .0668 ⇒ 6.7%
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(d) If you choose a tree at random, what is the probability it is
between 225 and 275 inches tall?
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Height of Fir Trees
Example
The height of fir trees in a certain forest follows a normal
distribution with µ = 240 and σ = 40 inches.
(d) If you choose a tree at random, what is the probability it is
between 225 and 275 inches tall?
z=
225 − 240
= −0.13
40
275 − 240
= 0.88
40
A = 0.0517 + 0.3106 = .3623
z=
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Approximating the Binomial Distribution
As we saw at the beginning of this section, the normal distribution
is shaped much like the binomial distribution. So much so that we
can use the normal distribution to approximate binomial
probabilities.
Normal Approximation to the Binomial Distribution
When n is large and p is close to 12 the normal distribution can be
used to approximate the binomial distribution with
µ = np
and
σ=
p
np(1 − p)
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Approximating the Binomial Distribution
As we saw at the beginning of this section, the normal distribution
is shaped much like the binomial distribution. So much so that we
can use the normal distribution to approximate binomial
probabilities.
Normal Approximation to the Binomial Distribution
When n is large and p is close to 12 the normal distribution can be
used to approximate the binomial distribution with
µ = np
and
σ=
p
np(1 − p)
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Approximating the Binomial Distribution
As we saw at the beginning of this section, the normal distribution
is shaped much like the binomial distribution. So much so that we
can use the normal distribution to approximate binomial
probabilities.
Normal Approximation to the Binomial Distribution
When n is large and p is close to 12 the normal distribution can be
used to approximate the binomial distribution with
µ = np
and
σ=
p
np(1 − p)
Why would we want to approximate binomial probabilities? Remember the president approval rating poll?
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Presidential Survey Question
Example
Suppose that the president has a job approval rating of 55%. If 30
people are surveyed, what is the probability that a majority
approve?
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Conclusion
Presidential Survey Question
Example
Suppose that the president has a job approval rating of 55%. If 30
people are surveyed, what is the probability that a majority
approve?
µ = 30(.55) = 16.5
p
σ = 30(.55)(.45) = 2.72
16 − 16.5
= −0.18
2.72
A = 0.0714 + 0.5000 = 0.5714
z=
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Presidential Survey Question, Part II
Example
In the same survey, find the probability that:
1
Between 18 and 25 people approve.
2
All but 4 people surveyed approve.
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Presidential Survey Question, Part II
Example
In the same survey, find the probability that:
1
Between 18 and 25 people approve.
2
All but 4 people surveyed approve.
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Presidential Survey Question, Part II
Example
In the same survey, find the probability that:
1
Between 18 and 25 people approve.
18 − 16.5
= 0.55
2.72
25 − 16.5
z=
= 3.13
2.72
A = 0.4990 − 0.2088 = .2902
z=
2
All but 4 people surveyed approve.
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Presidential Survey Question, Part II
Example
In the same survey, find the probability that:
1
Between 18 and 25 people approve.
2
All but 4 people surveyed approve.
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Presidential Survey Question, Part II
Example
In the same survey, find the probability that:
1
Between 18 and 25 people approve.
2
All but 4 people surveyed approve.
z=
4 − 16.5
= −4.60
2.72
A = 0.4990 + 0.5000 = .0.9990
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Outline
1
Bernoulli Probability and the Normal Distribution
2
Properties of the Normal Distribution
3
Examples
4
Conclusion
Examples
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Important Concepts
Things to Remember from Section 9-6
1
Properties of a normal distribution
2
Finding z-scores with z =
3
Finding areas using the standard normal curve table
4
Approximating binomial probabilities using the p
normal
distribution and in particular, µ = np and σ = np(1 − p)
z−µ
σ
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Important Concepts
Things to Remember from Section 9-6
1
Properties of a normal distribution
2
Finding z-scores with z =
3
Finding areas using the standard normal curve table
4
Approximating binomial probabilities using the p
normal
distribution and in particular, µ = np and σ = np(1 − p)
z−µ
σ
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Important Concepts
Things to Remember from Section 9-6
1
Properties of a normal distribution
2
Finding z-scores with z =
3
Finding areas using the standard normal curve table
4
Approximating binomial probabilities using the p
normal
distribution and in particular, µ = np and σ = np(1 − p)
z−µ
σ
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Important Concepts
Things to Remember from Section 9-6
1
Properties of a normal distribution
2
Finding z-scores with z =
3
Finding areas using the standard normal curve table
4
Approximating binomial probabilities using the p
normal
distribution and in particular, µ = np and σ = np(1 − p)
z−µ
σ
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Important Concepts
Things to Remember from Section 9-6
1
Properties of a normal distribution
2
Finding z-scores with z =
3
Finding areas using the standard normal curve table
4
Approximating binomial probabilities using the p
normal
distribution and in particular, µ = np and σ = np(1 − p)
z−µ
σ
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Next Time. . .
Next time we will review for Exam II. Exam II will cover sections
7-4 through 9-6 in your text.
For next time
Review sections 7-4 through 9-6
Conclusion
Bernoulli Probability and the Normal Distribution
Properties of the Normal Distribution
Examples
Next Time. . .
Next time we will review for Exam II. Exam II will cover sections
7-4 through 9-6 in your text.
For next time
Review sections 7-4 through 9-6
Conclusion