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Normal Curve
Normally Distributed Outcomes
Properties of Normal Curve
Standard Normal Curve
The Normal Distribution
Percentile
Probability for General Normal Distribution
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
The bell-shaped curve, as shown below, is call a
normal curve.
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



Examples of experiments that have normally
distributed outcomes:
1. Choose an individual at random and observe
his/her IQ.
2. Choose a 1-day-old infant and observe
his/her weight.
3. Choose a leaf at random from a particular
tree and observe its length.
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
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
A certain experiment has normally distributed
outcomes with mean equal to 1. Shade the
region corresponding to the probability that the
outcome
(a) lies between 1 and 3;
(b) lies between 0 and 2;
(c) is less than .5;
(d) is greater than 2.
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
The equation of
the normal
curve is
 1  x   
 

 2   
2
1
y
e
 2
where   3.1416 and e  2.7183.
The standard normal curve has   0 and   1.
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A(z) is the area
under the standard
normal curve to
the left of a
normally
distributed random
variable z.
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



Use the normal distribution table to determine
the area corresponding to
(a) z < -.5;
(b) 1< z < 2;
(c) z > 1.5.
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
(a) A(-.5) = .3085
(b) A(2) - A(1) = .9772 - 8413
= .1359
(c) 1 - A(1.5) = 1 - .9332
= .0668
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
If a score S is the pth percentile of a normal
distribution, then p% of all scores fall below S,
and (100 - p)% of all scores fall above S. The
pth percentile is written as zp.
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

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
What is the 95th percentile of the standard
normal distribution?
In the normal distribution, find the value of z
such that A(z) = .95.
A(1.65) = .9506
Therefore, z95 = 1.65.
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
If X is a random variable having a normal
distribution with mean  and standard
deviation , then
b 
a
b 
a 
Pr(a  X  b)  Pr 
Z

A

A















x 

 x 
and Pr( X  x)  Pr  Z 

A










where Z has the standard normal distribution
and A(z) is the area under that distribution to
the left of z.
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
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
Find the 95th percentile of infant birth weights if
infant birth weights are normally distributed
with
 = 7.75 and  = 1.25 pounds.
The value for the standard normal random
variable is z95 = 1.65.
Then x95 = 7.75 + (1.65)(1.25) = 9.81 pounds.
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
A normal curve is identified by its mean ()
and its standard deviation (). The standard
normal curve has = 0 and = 1. Areas of the
region under the standard normal curve can be
obtained with the aid of a table or graphing
calculator.
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
A random variable is said to be normally
distributed if the probability that an outcome
lies between a and b is the area of the region
under a normal curve from x = a to x = b. After
the numbers a and b are converted to standard
deviations from the mean, the sought-after
probability can be obtained as an area under
the standard normal curve.
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