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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