Download Normal Distribution Problem Step-by-Step Procedure

Normal Distribution Problem
Step-by-Step Procedure
Consider Normal Distribution Problem 2-37 on pages 62-63. We are given
the following information:
µ = 450, σ = 25
Find the following: P(X > 475) and P(460 < X < 470).
Z=
X −µ
σ
(a) Find P(X > 475)
Mean =450
X = 475
The formula to compute the Z value appears above. We apply that formula to
X − µ 475 − 450 25
the given data as follows: Z =
=
= =1
σ
25
25
A Z value of 1 means that X is located exactly one standard deviation to the
right of the mean. We need to the find the area of the normal curve that
corresponds to this Z value. Consult the Normal Distribution Table to find an
area of 0.84134 that corresponds to Z = 1. We want to find P(X > 475) so
this means we need the area to the right of X, which is:
1 - 0.84134 = 0.15866
Thus,
P(X > 475) = 0.16.
Normal Distribution Problem
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(b) Find P(460 < X < 470)
Mean=
450
X1=
460
X2=
470
This is a 2-step procedure where we find P(X < 470) and
P(X < 460) and then compute the difference. For reference, the
formula to compute the Z value appears to the right.
Z=
X −µ
σ
First, we apply that formula to find the Z value for X = 470 as follows:
X − µ 470 − 450 20
Z=
=
= = 0 .8
σ
25
25
We consult the Normal Distribution Table to find the area of 0.78814 that
corresponds to Z = 0.8.
Second, we apply that formula to find the Z value for X = 460 as follows:
X − µ 460 − 450 10
Z=
= = 0 .4
=
σ
25
25
We consult the Normal Distribution Table to find the area of 0.65542 that
corresponds to Z = 0.4.
Finally, we compute the difference between the 2 areas as follows:
P(460 < X < 470) = P(X < 460 < X) - P(X < 470) = 0.78814 - 0.65542 = 0.13272.
Thus, P(460 < X < 470) = 0.13.
Discussion: What are the relationships between Z, X, and the area under
the normal curve?
Normal Distribution Problem
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