Download PROBABILITY DISTRIBUTION

PROBABILITY
DISTRIBUTION
PROBABILITY DISTRIBUTIONS


We
usually
compare
our
observations in the studies with a
theoretical probability distribution
which
is
described
by
a
mathematical models.
Depending
variable is
probability
discrete or
on whether the random
discrete or continuous, the
distribution can be either
continuous.
Discrete probability distribution
1. Binomial distribution

Discrete probability distribution
2. Poisson distribution
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Used to determine the probability of
the rare events, when the average
number of successes is known.
Continuous probability distribution


we can only derive the probability of
the random variable if it is continuous
variable.
Continuous pr. Distributions include
(Normal test (z test), t test, F test
and Chi squared test)
The Normal Distribution
“Gaussian Distribution”


It is the most important distribution
in statistics
The parameters in this distribution
are the:
Population mean (µ) as a measure
of central tendency
 Population standard deviation (σ)
as a measure of dispersion
Mode
Median
The Normal Distribution
“Gaussian Distribution”
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


It is used for continuous variables
The curve is symmetric around the
mean
The total area under the curve equal
one
The mean, median, and the mode
are equal
The Normal Distribution
“Gaussian Distribution”


50% of the area under the curve is
on the right side of the curve and the
other 50% is on its left.
Not kurtotic and not skewed.
Skewness


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Refers to the degree of asymmetry of
the distribution of the variable.
It is skewed to the right (+ve) , it has
long tail to the right with few high
values ( the mean is larger than the
median).
It is skewed to the left(-ve) , it has
long tail to the left with few low values
( the mean is smaller than the median).
So the median is more stable than the
mean
Kurtosis



Refers to the ‘peakedness’ of the
distribution.
The flat curve is called platykurtotic.
The
peaked
curve
is
called
leptokurtotic.
Skewness
The Normal Distribution
“Gaussian Distribution”

µ±1σ
68% of the area

µ±2σ
95% of the area

µ±3σ
area
99.7% of the
The Normal Distribution
“Gaussian Distribution”

Different values of µ will shift the
graph of the distribution along the X
axis
The Normal Distribution
“Gaussian Distribution”

With fixed
(σ) any change in (µ)
will not change the shape of the
curve, but it will be shifted to:
the right ( when µ is increased)
or to the left (when µ is
decreased)
The Normal Distribution
“Gaussian Distribution”



Different values of (σ) determine
the degree of flatness or peakedness
of the graph of the distribution
When (σ) is increased the curve will
be more flat
When (σ) is decreased the curve will
be more peaked
The unit normal , or the Standard
normal distribution
X- µ
Z= ---------σ
Exercise 1
Find for a standard normal distribution
a) P(0< Z <1.2)
b) P(Z >1.2)
c)
P(-1.2< Z <1.2)
d) P(Z <-1.2 or Z >1.2)
e) P(Z <1.2)
f)
P(1.5 < Z <2.0)
Looking up probabilities in the standard
normal table
What is the area to the
left of Z=1.51 in a
standard normal curve?
Z=1.51
Z=1.51
Area is 93.45%
Exercise 2
If
µ of DBP of a population = 80 mmHg, and
σ2 =100(mmHg)2 .What is the probability
1)
2)
3)
4)
5)
6)
of selecting a man with DBP of:
P(75< X < 85)
P(60< X <100)
P(65< X <95)
P(X <60)
P(X >100)
P(90< X <100)
Exercise 3
If the weight of 6-years old boys are normally
distributed with µ =25 Kg, and
Find:
1.
P(20< X <25)
2.
P(X >28)
3.
P(X >22)
4.
P(X <22)
5.
P(X <28)
6.
P(26< X <29)
σ = 2 kg.
SAMPLING DISTRIBUTION
It is the distribution of all possible
values of a statistic (like mean),
computed from samples of the same
size randomly drawn from the same
population
STEPS IN CONSTRUCTING
SAMPLING DISTRIBUTION
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From a population of size (N), we
randomly draw all possible samples of size
(n)
From each sample we compute the
statistic of interest ( usually the mean)
STEPS IN CONSTRUCTING
SAMPLING DISTRIBUTION
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Make a table of the observed values
of the statistic and its corresponding
frequency
For any sampling distribution(of the
samples) we are interested in the
mean , variance, and the shape of
the curve.
DISTRIBUTION OF THE SAMPLE
MEAN
When sampling is from a normally
distributed population , the
distribution of the sampling mean
will posses the following properties:
 The distribution of the mean of the
_
samples (X) will be normal (Central
Limit Theorem).

DISTRIBUTION OF THE SAMPLE
MEAN
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The mean of the means of the
_
samples X will approximately be
equal to the mean of the
underlying population (µ) from
which these samples were drawn
DISTRIBUTION OF THE SAMPLE
MEAN
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The standard deviation of these
means will be σ/√n , n is the
number of values in the sample.
_
X- µ
Z=--------σ/√n
Exercise 4
If the cranial length of certain large
human population is normally
distributed with a mean =185.6 mm,
and standard deviation=12.7 mm.
What is the probability that a
random sample of size 10 from this
population will have a mean greater
than 190 mm?
Exercise
_
X- µ
190-185.6
Z=--------- =-------------=1.09
σ/√n
12.7 / √ 10
P for (Z=1.09) = 0. 8621
P (greater than 190 mm )
=1- 0.8621 =0.1379
Distribution of the difference of two
sample means
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EXERCISE 5
If the level of vitamin A in the liver
of two human populations is
normally distributed, the variance of
population 1 =19600 unit2, and of
population 2 =8100 unit2.
If there is no difference in population
means , what is the probability of
having a difference in means
between two samples (n1=15,
n2=10) drawn at random is equal or
greater than 50 unit.
DISTRIBUTION OF SAMPLE
PROPRTION
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EXERCISE 6
Suppose in a certain human
population , the prevalence of color
blindness is 8%. If we randomly
select 150 individuals from this
population, what is the probability
that the prevalence in the sample is
as great as 15%
DISTRIBUTION OF DIFFERENCE
BETWEEN TWO SAMPLE
PROPRTIONS
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EXERCISE 7
In a certain population of teenagers,
it is known that 10% of boys are
obese. If the same proportion of girls
in the population are obese, what is
the probability that a random sample
of 250 boys and 200 girls will yield a
difference in prevalence of 6%.
EXERCISE 8
It has been found that after a period
of training , the mean time required
for certain handicapped persons to
perform a particular task is 25
seconds, with a standard deviation of
5 seconds.
Assuming a normal distribution of
times, find the probability that a
sample of 25 individuals will yield a
mean of:
EXERCISE 8
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26 seconds or more
Between 24 and 27 seconds
26 seconds or less
Greater than 22 seconds
EXERCISE 9
In population A , the average annual
family expenditure on general health is
380 $, with a variance of 2899 $2. In
population B the comparative figures were
380 $, and 3250 $2.
A random sample of 40 families from
population A , and 35 families from
population B, showed average annual
expenditure on health of 346 $ and 300 $
,respectively.
Find the probability of such a difference or
higher?
EXERCISE10
A psychiatric social worker believes that in
both community A and community B , the
proportion of adolescents suffering from
some mental or emotional problem is
20%.
In a sample of 150 adolescents from
community A , 15 had mental or
emotional problem. In a sample of 100
from community B, the number was 16.
What is the probability of observing such
a difference or higher ?