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The probability that x could take any value in the range
between two given values a and b (a < b) is P(a ≤ x ≤ b)
P(a ≤ x ≤ b) is the
is
area colored in blue
area
under the normal
curve and between
the values
the
x = a and x = b
19 Three Important Areas under the Normal Curve 1. P (µ – σ ≤ x ≤ µ + σ) = 0.6826 So 68.26% of all possible observed values of x
are within (plus or minus) one standard
deviation of µ
2. P (µ – 2σ ≤ x ≤ µ + 2σ) = 0.9544 So 95.44% of all possible observed values of x
are within (plus or minus) two standard
deviations of µ
3. P (µ – 3σ ≤ x ≤ µ + 3σ) = 0.9973 So 99.73% of all possible observed values of x
are within (plus or minus) three standard
deviations of µ
20 Three Important Areas
under the Normal Curve (Visually)
The Empirical Rule for Normal Populations 21 The Standard Normal Distribution If x is normally distributed with mean µ and
standard deviation σ, then the random variable z x −µ
z=
σ
is normally distributed with mean 0 and standard
deviation 1; this normal is called the standard
normal distribution.
normal
22 The Standard Normal Distribution z measures the number of standard deviations that x is from the
mean µ
• The algebraic sign on z indicates on which side of µ is x
• z is positive if x > µ (x is to the right of µ on the number line)
• z is negative if x < µ (x is to the left of µ on the number line) 23 The Standard Normal Table Page 860861 The standard normal table is a table that lists the
The
cumulative areas under the standard normal curve. This table is very important. Always look at the accompanying figure for
Always
accompanying
guidance on how to use the table
guidance 24 The Standard Normal Table The values of z (accurate to the nearest
The
tenth) in the table range from 3.99 to 3.99
in increments of 0.01
in The areas under the normal curve to the left
The
of any value of z are given in the body of
the table
the 25 The Standardized Normal Table
0.9772 The Standardized
The
Normal table in the
textbook gives the
probability that z
Z
0 2.00
will be less than or
will
The row gives the value of z to the
equal to 2.00
second decimal point
Z
0.00 0.01 0.02 … 0.06 The column
column
0.0
shows the value 0.1
.
of z to the first
.
1.9.
decimal point
decimal
2.0 0.9750
. 9772 P(Z <= 2.00) = 0.9772, P(Z<=1.96)=0.9750
26 Find P (z ≤ 2)
Find Find the area listed in the table corresponding to a z
Find
value of 2.00
value Starting from the top of the far left column, go down
Starting
to “2.0”
to Read across the row z = 2.0 until under the colu...
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This note was uploaded on 09/16/2012 for the course 123 123 taught by Professor Vincent during the Spring '12 term at Ill. Chicago.
 Spring '12
 vincent

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