Normal Curve
„The
Probability & Statistics
bell-shaped curve, as shown below, is call a
normal curve.
The Normal Distribution
Normally Distributed Outcomes
„Examples
Properties of Normal Curve
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.
Example: Properties of Normal Curve
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.
Standard Normal Curve
„A
The equation of the
normal curve is
⎛ 1 ⎞⎛ x − µ ⎞
2
− ⎜ ⎟⎜
⎟
1
e ⎝ 2 ⎠⎝ σ ⎠
σ 2π
where π ≈ 3.1416 and e ≈ 2.7183.
y=
The standard normal curve has
µ = 0 and σ = 1.
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The Normal Distribution
Example: The Normal Distribution
„Use
A(z) is the area
under the standard
normal curve to the
left of a normally
distributed random
variable z.
the normal distribution table to
determine the area corresponding to
(a) z < -.5;
(b) 1< z < 2;
(c) z > 1.5.
Example
„
Find the value of z for which the area of the
shaded region under the standard normal curve
is .6915 to the right of z.
Example: Percentile
is the 95th percentile of the standard
normal distribution?
„What
65th
„What is the
percentile of the standard
normal distribution?
Percentile
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.
„If
Probability for
General Normal Distribution
If X is a random variable having a normal distribution
with mean µ and standard deviation σ , then
b−µ ⎞
⎛a−µ
⎛b−µ ⎞
⎛a−µ ⎞
Pr(a ≤ X ≤ b) = Pr ⎜
≤Z≤
⎟ = A⎜
⎟ − A⎜
⎟
σ
σ
σ
⎝
⎠
⎝
⎠
⎝ σ ⎠
x−µ ⎞
⎛
⎛ x−µ ⎞
and Pr( X ≤ x) = Pr ⎜ Z ≤
= A⎜
⎟
σ ⎟⎠
⎝
⎝ σ ⎠
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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Example
Example: Probability Normal Distribution
„
Find the 95th percentile of infant birth weights if
infant birth weights are normally distributed with
µ = 7.75 and σ = 1.25 pounds.
„
Suppose that the height (at the shoulder) of
adult African bull bush elephants is normally
distributed with µ = 3.3 meters and σ = .2
meter. The elephant on display at the
Smithsonian Institution has height 4 meters and
is the largest elephant on record.
What is the probability that an adult African bull
bush elephant has height 4 meters or more?
Summary
A normal curve is identified by its mean ( µ )
and its standard deviation (σ ). The standard
normal curve has µ = 0 and σ = 1.
¾Areas of the region under the standard
normal curve can be obtained with the aid of a
table or graphing calculator.
¾
Summary
¾A
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.
¾After 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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