Download normal distribution

Lecture 7
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To understand what a Normal Distribution is
To know how to use the Normal Distribution
table
To compute probabilities of events by using
the Normal Distribution table
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(a) Many distributions in biology, geography,
economics, and demographic studies are
approximately normal.
(b) In industrial production, the mass, size and
other features of mass-produced products
will follow an approximately normal
distribution. This is of fundamental
importance in quality control.
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(c) Sets of random errors are approximately
normally distributed. This result was much
studied by Gauss and the normal distribution is
sometimes called the ‘Gaussian distribution’ or
the ‘error curve’.
(d) It can be used as an approximation to the
binomial and Poisson distributions (under
certain conditions) and to other distributions.
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A very important
distribution.
continuous
probability
Many natural and economic phenomena tend
to be approximately normally distributed.
Examples:
heights,
weights,
intelligent
quotients, physically productions.
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Normal Distribution Curve
Median
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Symmetrical & bell-shaped.
It is uni-modal.
The mean, median & mode all lie at the
centre of distribution.
The two tails never touch the horizontal axis,
although they come close to it.
The area under the curve is 1.
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Since the normal distribution shows a
distribution of continuous variables, we
cannot pinpoint the probability of a
particular point on the distribution.
We can, however, compute the probabilities
of any intervals on the distribution.
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There is a unique relationship between the
areas under a normal curve and its standard
deviation, σ .
If we draw a vertical line at the centre of the
curve, and measure 1σ away from the curve
on either side, then the area enclosed
between the two vertical lines at the 1σ and
the curve is about 68% of the total area
under the whole curve.
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Statisticians can compute the area of any
intervals under the curve in units of σ.
A table of such values has, in fact, been
worked out.
As long as we can find out the distance
between the centre of the curve and any point
in terms of the units of σ , we can obtain the
required area.
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Example
In a normal distribution which has a mean of
100 and a standard deviation of 20, what is
the area between the centre of the curve and
a variable, 130 on the horizontal axis.
Solution
Z = (130 – 100)/20 =1.5
Then we look for Z = 1.5 from table which
corresponds to 0.4332
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The key to working with the Normal distribution is to
transform the normal variable X into the standard normal
variable Z by the operation
Z=
Where
x

x = Value of a random variable
 = mean of a normal distribution
 = standard deviation of the distribution
so that the new mean and standard deviation is 0 and 1
respectively.
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Example
For a normal probability distribution with
mean µ = 5 and σ =2, find the following
probabilities
a) P(X < 6.5)
b) P(X > 6.5)
c) P(4.5 < X < 6.5)
d) P(6.5 < X < 8.0)
Example
If x is normally distributed with μ = 20.0 and
σ=4.0, determine the following:
a) P( x ≥ 20.0)
b) P(16.0 ≤ x ≤ 24.0)
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Example
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It has been reported that the average hotel check-in
time, from curbside to delivery of bags into the room, is
12.1 minutes. An Li has just left the cab that brought
her to her hotel.
Assuming a normal distribution with a standard
deviation of 2.0 minutes, what is the probability that
the time required for An Li and her bags to get to the
room will be:
1.
Greater than 14.1 minutes?
2.
Between 10.1 and 14.1 minutes?
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Explain Normal Distribution.
Use the Standard Normal Distribtuon Table.
Compute probabilities of events using the
standard normal distribution table.