Surfstat.australia: an online text in introductory Statistics
SUMMARISING AND PRESENTING DATA
NORMAL DISTRIBUTIONS
Example: z < 1.05
To find the relative frequency of getting a value of Z less than 1.05 look
at Table 1. Go down the first column to the row with Z = 1.0 and then across
to the column headed .05 (i.e. Z = 1.05). The value in the corresponding
cell of the table is the required relative frequency, 0.8531.
Thus the relative frequency that Z < 1.05, where Z ~ N(0, 1), is 0.8531.
Other relative frequencies can be found using the symmetry of the curve
and the fact that the total area is 1.
Example: z > 1.05
The relative frequency with which Z > 1.05 is 1 -
0.8531 = 0.1469
Example: -1.05 <= z <= 1.05
The proportion of observations for which (-1.05
Z1.05) is equivalent to
(Z1.05) - (Z -1.05)
P(Z1.05) - (Z
1.05)
P(Z1.05) - [1 - (Z1.05)]
or 2 x P(Z1.05) - 1 = 2 x 0.8531 - 1
= 0.7062
Exercise
Check that the relative frequencies of each of the following events is,
approximately
Relative frequencies for Normal distributions other than the standard
Normal distribution N(0,1) are obtained by using the formula
Example - X ~ N(4, 9), X < 5
Example - Soft drink bottle filler
A filling machine is used to fill soft drink bottles. The bottles are
supposed to contain 300 mls. In fact the quantities vary according to
the Normal distribution with expected value of µ = 298ml and
standard deviation s = 3ml.
What proportion of bottles contain less than 295 mls?
Let X denote the quantity in an individual bottle. We are told 
We want the relative frequency of X < 295.
i.e. about 16% of bottles would have less than 295 ml.