Download Definition: A normal distribution is a distribution described by a

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Definition: A normal distribution is a distribution
described by a normal density curve. A normal distribution is completely specified by two numbers, µ and
σ.
The 68-95-99.7 Rule: In the normal distribution
N (µ, σ):
(I) Approximately 68% of the observations fall within
σ of the mean µ.
(II) Approximately 95% of the observations fall within
2σ of the mean µ.
(III) Approximately 99.7% of the observations fall within
3σ of the mean µ.
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(A) Corresponding to the case (I) for N (0, 1), we have
the following graph
0.4
density curve
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−3
−2
−1
0
1
2
3
The area is equal to 68%
(B) Corresponding to the case (II) for N (0, 1), we have
the following graph
0.4
density curve
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−3
−2
−1
0
1
2
The area is equal to 95%
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(C) Corresponding to the case (III) for N (0, 1), we have
the following graph
0.4
density curve
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−4
−3
−2
−1
0
1
2
3
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The area is equal to 99.7%
Example: The length of human pregnancies from conception to birth varies according to a distribution that
is approximately normal with mean 266 days and standard deviation 16 days. Use the 68 − 95 − 99.7 rule to
answer the following questions.
(a) Between what values do the lengths of almost all
(99.7%) pregnancies fall?
(b) How short are the shortest 2.5% of all pregnancies?
Solution: (a) According to (III), within 3 standard deviations of the mean,i.e.
[218 = (266 − 3 × 16),
314 = (266 + 3 × 16)]
contains 99.7% of the observations.
(b) From the graph
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0.025
density curve
0.02
0.015
0.01
0.005
0
200
220
240
260
280
300
320
The area is equal to 95%
and (II),
[234 = (266 − 2 × 16),
298 = (266 + 2 × 16)]
contains 95% of the observations. Since normal distribution is symmetric, there are 2.5% of observations are
less than 234 days.
Definition: A standard normal distribution is the
N (0, 1). If a (random) variable X has a normal distribution N (µ, σ), then
Z=
X −µ
σ
is a standardized variable which has distribution N (0, 1).
Cumulative proportion or probability P(X ≤ x):
The cumulative proportion for a given value x in the
distribution is the proportion of observations in the
distribution that lie at or below x. In other words, it
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is the probability to observe a value of the variable X
less than or equal to x and this is denoted by
P(X ≤ x).
The following figure show the cumulative proportion
less than or equal to −1 for Z ∼ N (0, 1).
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density curve
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
−4
−3
−2
−1
0
1
2
3
The area less than or equal to −1
Using the standard normal table: We can find
P(Z ≤ x)
by the standard normal table on page 684-685, Table
A.
Example:
P(Z ≤ −1.85) = 0.0322
and
P(Z ≤ 1.64) = 0.9495.
Generally if X ∼ N (µ, σ), then the transformation
X −µ
Z=
σ
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define a new variable which has a standard normal
distribution.
Example: If X ∼ N (0.5, 0.5), find P(X ≤ 1).
Solution: Since Z = X−0.5
0.5 ∼ N (0, 1), we have
X − 0.5 1 − 0.5
≤
) = P(Z ≤ 1) = 0.8413.
0.5
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Remark: If variable X has a distribution which is
specified by a density curve, then for any constant a,
P(X = a) = 0. Therfore,
P(X ≤ 1) = P(
P(X ≤ a) = P(X < a) + P(X = a) = P(X < a).