Transcript Normal distribution.

Theoretical distributions:
the Normal distribution
The Aim
By the end of this lecture, the
students will be aware of the Normal
distribution
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The Goals
-Define the terms: probability, conditional probability
-Distinguish between the subjective, frequentist and a
priori approaches to calculating a probability
-Define the addition and multiplication rules of probability
-Define the terms: random variable, probability
distribution, parameter, statistic, probability density
function
-Distinguish between a discrete and continuous
probability distribution and list the properties of each
-List the properties of the Normal and the Standard
Normal distributions
-Define a Standardized Normal Deviate (SND)
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Normal distribution
-Probability
-Rules of probabiliyty
-Probability distribution
-Normal (Gaussian) distibution
-Standard normal distribution
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-In previous lectures we showed how to create an
empirical frequency distribution of the observed data.
-This contrasts with a theoretical probability
distribution which is described by a mathematical
model.
-When our empirical distribution approximates a
particular probability distribution, we can use our
theoretical knowledge of that distribution to answer
questions about the data.
-This often requires the evaluation of probabilities.
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Understanding probability
-Probability measures uncertainty; it lies at the
heart of statistical theory.
-A probability measures the chance of a given
event occurring. It is a number that takes a
value from zero to one. If it is equal to zero,
then the event can not occur. If it is equal to
one, then the event must occur.
-The probability of the complementary event
(the event not occurring) is one minus the
probability of the event occurring.
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Understanding probability
We can calculate a probability using various approaches.
•
Subjective - our personal degree of belief that the event
will occur (e.g. that the world will come to an end in the year
2050).
•
Frequentist - the proportion of times the event would
occur if we were to repeat the experiment a large number of
times (e.g. the number of times we would get a 'head' if we
tossed a fair coin 1000 times).
•
A priori - this requires knowledge of the theoretical
model, called the probability distribution, which describes the
probabilities of all possible outcomes of the 'experiment'.
For example, genetic theory allows us to describe the
probability distribution for eye colour in a baby born to a blueeyed woman and brown-eyed man by specifying all possible
genotypes of eye colour in the baby and their probabilities.
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The rules of probability
We can use the rules of probability to add and multiply probabilities.
-The addition rule - if two events, A and B, are mutually exclusive
(i.e. each event precludes the other), then the probability that either
one or the other occurs is equal to the sum of their probabilities.
Prob(A or B) = Prob(A) + Prob(B)
For example, if the probabilities that an adult patient in a particular
dental practice has
-no missing teeth (0.67),
-some missing teeth (0.24) or is
-edentulous (i.e. has no teeth) (0.09) then
the probability that a patient has some teeth is 0.67 + 0.24 = 0.91
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The rules of probability
-The multiplication rule - if two events, A and B, are independent
(i.e. the occurrence of one event is not contingent on the other),
then the probability that both events occur is equal to the product
of the probability of each:
Prob(A and B) = Prob(A) x Prob(B)
For example,
if two unrelated patients are waiting in the dentist's surgery,
the probability that both of them have no missing teeth is
0.67 X 0.67 = 0.45
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Probability distributions: the theory
-A random variable is a quantity that can
take any one of a set of mutually
exclusive values with a given probability.
-A probability distribution shows the
probabilities of all possible values of the
random variable.
-It is a theoretical distribution that is
expressed mathematically, and has a
mean and variance that are analogous to
those of an empirical distribution.
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Probability distributions: the theory
-Each probability distribution is defined by
certain parameters which are summary
measures (e.g. mean, variance)
characterizing that distribution (i.e. knowledge
of them allows the distribution to be fully
described).
-These parameters are estimated in the
sample by relevant statistics.
-Depending on whether the random variable
is discrete or continuous, the probability
distribution can be either discrete or
continuous.
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Probability distributions: the theory
Discrete (e.g. Binomial and Poisson)
- We can drive probabilities corresponding
to every possible value of the random
variable.
- The sum of all such probabilities is one.
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Probability distributions: the theory
Continuous (e.g. Normal, Chi-squared, t and F)
-We can only derive the probability of the random variable, x, taking
values in certain ranges (because there are infinitely many values of x).
-If the horizontal axis represents the values of x, we can draw a curve
from the equation of the distribution (the probability density function); it
resembles an empirical relative frequency distribution.
-The total area under the curve is one; this area represents the
probability of all possible events.
-The probability that x lies between two limits is equal to the area under
the curve between these values (Fig. 7.1).
-For convenience, tables have been produced to enable us to evaluate
probabilities of interest for commonly used continuous probability
distributions.
-These are particularly useful in the context of confidence intervals and
hypothesis testing.
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Probability distributions: the theory
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The Normal (Gaussian) distribution
• One of the most important distributions in statistics
is the Normal distribution. Its probability density
function (Fig. 7.2) is:
– completely described by two parameters,
the mean (µ) and the variance (σ2);
– bell-shaped (unimodal);
– symmetrical about its mean;
– shifted to the right if the mean is increased and to the
left if the mean is decreased (assuming constant
variance);
– flattened as the variance is increased but becomes
more peaked as the variance is decreased (for a fixed
mean).
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The Normal (Gaussian) distribution
Additional properties are that:
-the mean and median of a Normal distribution are equal;
-the probability (Fig. 7.3a) that a Normally distributed random variable, x,
with mean, µ, and standard deviation, σ , lies between
(µ - σ) and (µ+ σ) is 0.68
(µ - l.96σ) and (µ + 1.96σ) is 0.95
(µ - 2.58σ) and (µ+ 2.58σ) is 0.99
(These intervals may be used to define reference intervals)
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68-95-99 Rule
68% of
the data
95% of the data
99% of the data
Converting to a Standard
Normal Distribution
z=
x–

Figure 6-12
Copyright © 2007
Pearson Education, Inc
The Normal Distribution
f(X)
Changing μ shifts the
distribution left or right.


Changing σ increases
or decreases the
spread.
X
The Normal Distribution:
as mathematical function (pdf)
f ( x) 
1
 2
Note constants:
=3.14159
e=2.71828
1 x 2
 (
)
2

e
This is a bell shaped
curve with different
centers and spreads
depending on  and 
Properties of Normal Distributions
A continuous random variable has an infinite number of possible values that
can be represented by an interval on the number line.
Hours spent studying in a day
0
3
6
9
12
15
18
21
24
The time spent
studying can be any
number between 0
and 24.
The probability distribution of a continuous random variable is called a
continuous probability distribution.
Properties of Normal Distributions
The most important probability distribution in statistics is the normal
distribution.
Normal curve
x
A normal distribution is a continuous probability distribution for a random
variable, x. The graph of a normal distribution is called the normal curve.
The Standard Normal distribution
There are infinitely many Normal distributions
depending on the values of µ and σ .
The Standard Normal distribution (Fig. 7.3b) is a
particular Normal distribution for which probabilities
have been tabulated.
-The Standard Normal distribution has a mean
of zero and a variance of one.
-If the random variable x has a Normal
distribution with mean µ and variance σ2, then the
Standardized Normal Deviate (SND),
z = x - µ, is a random variable that has a
Standard Normal distribution.
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The Standard Normal Distribution
(Z)
All normal distributions can be converted into
the standard normal curve by subtracting the
mean and dividing by the standard deviation:
Z
X 

Somebody calculated all the integrals for the standard
normal and put them in a table! So we never have to
integrate!
Even better, computers now do all the integration.
Comparing X and Z units
100
0
200
2.0
X
Z
( = 100,  = 50)
( = 0,  = 1)
The Standard Normal Distribution
The standard normal distribution is a normal distribution
with a mean of 0 and a standard deviation of 1.
The horizontal scale
corresponds to z-scores.
z
3
2
1
0
1
2
Any value can be transformed into a z-score by using the formula
z=
Value - Mean
x -μ.
=
Standard deviation
σ
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The Standard Normal Distribution
If each data value of a normally distributed random
variable x is transformed into a z-score, the result will be
the standard normal distribution.
The area that falls in the interval under the
nonstandard normal curve (the x-values)
is the same as the area under the
standard normal curve (within the
corresponding z-boundaries).
z
3
2
1
0
1
2
3
After the formula is used to transform an x-value into a z-score, the Standard
Normal Table in Appendix B is used to find the cumulative area under the
curve.
The Standard Normal Table
Properties of the Standard Normal Distribution
1. The cumulative area is close to 0 for z-scores close to z = 3.49.
2. The cumulative area increases as the z-scores increase.
3. The cumulative area for z = 0 is 0.5000.
4. The cumulative area is close to 1 for z-scores close to z = 3.49
Area is close to 1.
Area is close to 0.
z = 3.49
3
z
2
1
0
1
z=0
Area is 0.5000.
2
3
z = 3.49
The Standard Normal Table
Example:
Find the cumulative area that corresponds to a z-score
of 2.71.
Appendix B: Standard Normal Table
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
.5000
.5040
.5080
.5120
.5160
.5199
.5239
.5279
.5319
.5359
0.1
.5398
.5438
.5478
.5517
.5557
.5596
.5636
.5675
.5714
.5753
0.2
.5793
.5832
.5871
.5910
.5948
.5987
.6026
.6064
.6103
.6141
2.6
.9953
.9955
.9956
.9957
.9959
.9960
.9961
.9962
.9963
.9964
2.7
.9965
.9966
.9967
.9968
.9969
.9970
.9971
.9972
.9973
.9974
2.8
.9974
.9975
.9976
.9977
.9977
.9978
.9979
.9979
.9980
.9981
Find the area by finding 2.7 in the left hand column, and
then moving across the row to the column under 0.01.
The area to the left of z = 2.71 is 0.9966.
The Standard Normal Table
Example:
Find the cumulative area that corresponds to a z-score
of 0.25.
Appendix B: Standard Normal Table
z
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
3.4
.0002
.0003
.0003
.0003
.0003
.0003
.0003
.0003
.0003
.0003
3.3
.0003
.0004
.0004
.0004
.0004
.0004
.0004
.0005
.0005
.0005
0.3
.3483
.3520
.3557
.3594
.3632
.3669
.3707
.3745
.3783
.3821
0.2
.3859
.3897
.3936
.3974
.4013
.4052
.4090
.4129
.4168
.4207
0.1
.4247
.4286
.4325
.4364
.4404
.4443
.4483
.4522
.4562
.4602
0.0
.4641
.4681
.4724
.4761
.4801
.4840
.4880
.4920
.4960
.5000
Find the area by finding 0.2 in the left hand column, and
then moving across the row to the column under 0.05.
The area to the left of z = 0.25 is 0.4013
The Standard Normal Table
What is the area to the
left of Z=1.51 in a
standard normal curve?
Area is 93.45%
Z=1.51
Z=1.51
•Examples
Ex. 3: Estimating a Probability for a Normal Curve
• Adult IQ scores are normally distributed with
•  = 100 and  = 15.
• Estimate the probability that a randomly chosen adult has an IQ
between 70 and 115.
Solution: Draw the curve.
Ex. 3: Estimating a Probability for a Normal Curve
• Adult IQ scores
are normally
distributed with 
= 100 and  =
15. Estimate the
probability that a
randomly
chosen adult
has an IQ
between 70 and
115.
Using the Empirical Rule, the area under the
normal curve between these two values is:
Solution: Draw the
curve.
Area = .135 + .34 + .34 = .815 So the probability
the adult has an IQ between 70 and 115 is about
.815
70
85
100
115
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Guidelines for Finding Areas
Example:
Find the area under the standard normal
curve to the left of z = 2.33.
Always draw the
curve!
z
2.33
0
From the Standard Normal Table, the area is equal to 0.0099.
Guidelines for Finding Areas
Example:
Find the area under the standard normal
curve to the right of z = 0.94.
Always draw the
curve!
0.8264
1  0.8264 = 0.1736
z
0
0.94
From the Standard Normal Table, the area is equal to 0.1736.
Guidelines for Finding Areas
Example:
Find the area under the standard normal
curve between z = 1.98 and z = 1.07.
Always draw the
curve!
0.8577
0.8577  0.0239 = 0.8338
0.0239
z
1.98
0
1.07
From the Standard Normal Table, the area is equal to 0.8338.
Probability and Normal Distributions
If a random variable, x, is normally distributed, you
can find the probability that x will fall in a given
interval by calculating the area under the normal
curve for that interval.
μ = 10
σ=5
P(x < 15)
x
μ =10
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Probability and Normal Distributions
Example:
The average on a statistics test was 78 with a standard
deviation of 8. If the test scores are normally distributed,
find the probability that a student receives a test score
less than 90.
μ = 78
σ=8
z  x - μ = 90 - 78
σ
8
= 1.5
P(x < 90)
x
μ =78
90
z
μ =0
P(x < 90) = P(z < 1.5) = 0.9332
?
1.5
The probability that a
student receives a test
score less than 90 is
0.9332.
Probability and Normal Distributions
Example:
The average on a statistics test was 78 with a standard
deviation of 8. If the test scores are normally distributed,
find the probability that a student receives a test score
greater than than 85.
μ = 78
σ=8
P(x > 85)
z = x - μ = 85 - 78
σ
8
= 0.875  0.88
x
μ =78 85
z
μ =0 0.88
?
The probability that a
student receives a test
score greater than 85 is
0.1894.
P(x > 85) = P(z > 0.88) = 1  P(z < 0.88) = 1  0.8106 = 0.1894
Probability and Normal Distributions
Example:
The average on a statistics test was 78 with a standard
deviation of 8. If the test scores are normally distributed,
find the probability that a student receives a test score
between 60 and 80.
z = x - μ = 60 - 78 = -2.25
σ
8
z 2  x - μ = 80 - 78
σ
8
1
P(60 < x < 80)
μ = 78
σ=8
x
60
μ =78 80
2.25
μ =0 0.25
?
?
z
= 0.25
The probability that a
student receives a test
score between 60 and 80
is 0.5865.
P(60 < x < 80) = P(2.25 < z < 0.25) = P(z < 0.25)  P(z < 2.25)
= 0.5987  0.0122 = 0.5865
Summary
Normal distribution
-Probability
-Rules of probabiliyty
-Probability distribution
-Normal (Gaussian) distibution
-Standard normal distribution
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