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Section 7.6: The Normal Distribution.
MTH 245: Mathematics for Management, Life,
and Social Sciences
F. Patricia Medina
Department of Mathematics. Oregon State University
November 13, 2014
Section 7.6
Section 7.6: The Normal Distribution.
The Normal Distribution.
Figure : Abraham DeMoivre
Section 7.6: The Normal Distribution.
The normal curve
Abraham DeMoivre proved that areas under the curve
1
2
fz (z) = √ e−z /2 , can be used to estimate
2π


X − n 12
 , where X is a random binomial random
P a ≤ q
1
1
n 2 2 ≤b
variable with a large n and p = 12 .
The “curve” is called normal curve. We will take a look into
continuous probability by studying experiments with
normally distributed outcomes. For instance, we might consider
choosing a newborn and observing his or her weight, chose a college
student on campus and observe his or her height, etc. Associated to
each experiment is the normal curve, a bell-shaped curve.
Section 7.6: The Normal Distribution.
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Section 7.6: The Normal Distribution.
Probability for a normal distribution
The probability that the value of the random variable, X, lies between
two values, a and b, is the fraction of the area under the normal curve
that lies between x = a and x = b, we denote it by P(a ≤ X ≤ b).
The total area under the normal curve is always 1.
A normal curve is completely described by its mean µ and standard
deviation σ .
G
iven µ and σ we write down the equation of the associated normal
curve as
x−µ 2
1
1
y = √ e−( 2 )( σ ) .
σ 2π
The standard normal curve has mean µ = 0 and standard deviation
(s.d.) σ = 1.
Section 7.6: The Normal Distribution.
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Section 7.6: The Normal Distribution.
Let Z be a random variable having the standard normal distribution.
Let z be any number, A(z) denotes the area under the standard
normal curve to the left of z
If X is a random variable having a normal distribution with mean µ
and standard deviation σ , then
a−µ
b−µ
b−µ
a−µ
P(a ≤ X ≤ b) = P
≤Z≤
=A
−A
σ
σ
σ
σ
Section 7.6: The Normal Distribution.
and
x−µ
x−µ
P(X ≤ x) = P 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.
z-Scores allow us to use a table to find the amount of area that is to
the left of a given value,x , in a normal distribution.
value of x − mean
x−µ
=
.
standard deviation
σ
The z-score should be interpreted as the number of standard
deviations above/below the mean.
Z − score =
Section 7.6: The Normal Distribution.
Appendix A
Section 7.6: The Normal Distribution.
Example 1
For the case where x = 38.4, µ = 22.5, and σ = 6.2
a) Find the z-score.
b) What percent of the area of the normal curve lies below
x = 38.4?
Section 7.6: The Normal Distribution.
Example 2
Find the area under the normal curve with µ = 7, σ = 2 from x = 6 to
x = 10. This represents P(6 ≤ X ≤ 10) for a random variable X
having the given normal distribution.
Section 7.6: The Normal Distribution.
Definition 3
If a score S is in the pth percentile of a normal distribution, then p%
of all scores fall bellow S, and (100 − p)% of all scores fall above S.
Example 4
What is the 90th percentile of the standard normal distribution?
Section 7.6: The Normal Distribution.
SAT scores
Example 5
Assume that SAT verbal scores for a first-year class at a university are
normally distributed with mean 520 and standard deviation 75.
(a) The top 10% of the students are placed into the honors
program for English. What is the lowest score for
admittance into the honors program?
(b) What is the range of the middle 90% of the SAT verbal
scores at the university?
(c) Find the 98th percentile of the SAT verbal scores.
Section 7.6: The Normal Distribution.
Example 6
The lifetime of a certain brand of tires ins normally distributed with
mean µ = 30, 000 miles and standard deviation σ = 5000. The
company has decided to issue a warranty for the tires but does not
want to replace more than 2% of the tires that it sells. At what
mileage should the warranty expires?