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headed by “.00”
headed The area is in the cell that is the intersection of this
The
row with this column
row As listed in the table, the area is 0.9772, so
As
P (z ≤ 2) = 0.9772 27 Some Areas under the Standard Normal Curve 28 Calculating P (2.53 ≤ z ≤ 2.53) First, find P(z ≤ 2.53) Go to the table of areas under the standard normal
curve Go down leftmost column for z = 2.5 Go across the row 2.5 to the column headed by .03 The cumulative area to the value of z = 2.53 is the
value contained in the cell that is the intersection of
the 2.5 row and the .03 column The table value for the area is 0.9943 29 Calculating P (2.53 ≤ z ≤ 2.53) From last slide, P ( z ≤ 2.53)=0.9943
From
By symmetry of the normal curve,
By
P(z ≤ 2.53)=10.9943=0.0057
P(z
Then P (2.53 ≤ z ≤ 2.53)
Then
= P ( z ≤ 2.53)P(z ≤ 2.53)
= 0.9943 0.0057 = 0.9886
0.9943
Alternative:
P (0 ≤ z ≤ 2.53)=0.99430.5=0.4943
P (2.53 ≤ z ≤ 2.53)=0.4943+0.4943=0.9886 30 Calculating P(z ≥ 1)
An example of finding
An
tail areas
tail
• Shown is finding the
Shown
righthand tail area for
z ≥ 1.00
• Equivalent to the
Equivalent
lefthand tail area
for z ≤ 1.00
31 Calculating P (z ≥ 1)
An example of finding the area under the standard
normal curve to the right of a negative z value
• Shown is finding the under the standard normal
for z ≥ 1 32 Finding Normal Probabilities
General procedure:
1. Formulate the problem in terms of x values
1.
2. Calculate the corresponding z values, and restate the
2.
problem in terms of these z values
3. Find the required areas under the standard normal curve
3.
by using the table
by
Note: It is always useful to draw a picture showing the
Note:
required areas before using the normal table
required
33 Finding Normal Probabilities Probability is the area under the curve!
f(X) X is normal random variable with mean µ P (a ≤ X ≤ b) and s.d. σ
Z follows
standard normal
distribution a b X P (a ≤ X ≤ b)
a−µ
b−µ
= P(
≤Z ≤
)
σ
σ
a b z
34 Example 5.2
Example The Car Mileage Case Want the probability that the mileage of a randomly
selected midsize car will be between 32 and 35 mpg
• Let x be the random variable of mileage of midsize
cars, in mpg
• Want P (32 ≤ x ≤ 35 mpg)
Given:x is normally distributed
µ = 33 mpg
σ = 0.7 mpg 35 The Car Mileage Case #2
Procedure (from previous slide):
1. Formulate the problem in terms of x (as above)
1.
2. Restate in terms of corresponding z values (see
2.
next slide)
next
3. Find the indicated area under the...
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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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