Transcript Anatomy: x to z Conversion

ANATOMY OF CONVERTING AN X VALUE TO A Z VALUE
If x is a normally (or approximately normally) distributed random variable, with mean other than 0 and/or standard deviation other than 1, the value(s) of x must be
converted to the Standard Normal Distribution in order to determine probabilities. The formula used to convert x to a z-value will depend upon whether an
individual unit or a sample from a population is being considered.
Individual unit from a population:
When considering an individual unit from population with mean, ,
and standard deviation, , use the formula below to convert a value of
x from a nonstandard normal distribution, to a z-value from the
Standard Normal Distribution.
z
x
z

EXAMPLE: Given that the scores of students on the ACT college entrance
examination in a recent year were normally distributed with mean,  = 18.6
and standard deviation  = 5.9, What is the probability that a single student
randomly chosen from all those taking the test scores 21 or higher?
In this case we are considering a single unit from the population.
In order to find P(x > 21) we will need to convert the x-value to a z-value
from the Standard Normal Distribution as is shown below. Remember to
draw a picture of what you are looking for.
z
x

Sample from a population:
When considering a sample from a population with mean, , and standard
deviation , , we will use the formula below to convert the mean of the sample
(x-bar) from a nonstandard normal distribution, to a z-value from the Standard
Normal Distribution.
__
EXAMPLE: Given that the scores of students on the ACT college entrance
examination in a recent year were normally distributed with mean,  = 18.6 and
standard deviation  = 5.9, what is the probability that the mean score obtained
from a sample of 50 students who took the test is 21 or higher?
In this case we are considering a sample from the population.
__
In order to find P( x  21) we will need to convert the value of x to a z-value
from the Standard Normal Distribution as is shown below. Remember to draw a
picture of what you are looking for.
z
We can now go to the Standard Normal Table to find P(z > .41) = .3409.
( Obtaining this probability will vary depending on the table format.)
P( x  21)  P( z  .41)  .3409

n
__
21  18.6 2.4


 .41
5.9
5. 9
x 
x 

n

21  18.6 2.4
2.4


 2.87
5.9
5.9 .835
7.07
50
We can now go to the Standard Normal Table to find P ( z  2.87)  .0021.
( Obtaining this probability will vary depending on the table format.)
__
P( x  21)  P( z  2.87)  .0021
Area of interest
to right of 2.87
Area of interest
to right of .41
0
.41
0
2.87