Page 49 - Basic Statistics

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density(rnorm(100000, mean = 0, sd = 1), width = 10)$y 0.15 0.10 0.05

0.0
-10 -5 0 5 10
density(rnorm (100000, m ean = 0, sd = 1), width = 10)$x

____!______!______!______!______!______!______!____ X
 -3  -2  -1   +1  +2  +3
! ! ! ! ! ! !
____!______!______!______!______!______!______!____ Z
-3 -2 -1 0 1 2 3

Figure 3.4 Normal random variable X
and transformed into standard normal distribution Z

Distribution of the origin and distribution of the results of the
transformation is illustrated in Figure 3.4. Because all values of X that falls

between x and x have the equivalent values of z between z and z , then the
1
2
2
1
area under the normal curve X between x = x and x = x in Figure 3.1 is equal
1
2
to the area under the standard normal curve between the values of Z the results
of the transformation z = z and z = z .Thus
2
1
P( x < X < x ) = P( z < Z < z )
2
2
1
1

3.4 USING THE STANDARD NORMAL DISTRIBUTION TABLE

In using the standard normal table, we reduce some tables on the area of
the normal curve into only one, which is derived from the standard normal

distribution. Table A.1 lists the area under the standard normal curve which is

the value of P (0 < Z < z) for various values of z from 0 to 5.49. To illustrate the

use of this table, let us count the probability that Z takes values between 0 to

~~* CHAPTER 3 NORMAL PROBABILITY DISTRIBUTION *~~