Normal Distribution Problem Step-by-Step Solution
[Pages:2]Normal Distribution Problem Step-by-Step Procedure
Consider Normal Distribution Problem 2-37 on pages 62-63. We are given the following information:
? = 450, = 25
Find the following: P(X > 475) and P(460 < X < 470).
Z = X -?
(a) Find P(X > 475)
Mean =450 X = 475
The formula to compute the Z value appears above. We apply that formula to
the given data as follows: Z = X - ? = 475 - 450 = 25 =1
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=
X1=
X2=
450
460
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 -?
First, we apply that formula to find the Z value for X = 470 as follows:
Z = X - ? = 470 - 450 = 20 = 0.8
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:
Z = X - ? = 460 - 450 = 10 = 0.4
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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