AP Statistics: Section 2.2 B
The Standard Normal Distribution
As the 68-95-99.7 Rule suggests, all Normal
distributions share many common
properties. In fact, all Normal distributions
are the same if we measure in units of size
 with the mean  as center.
Changing to these units requires us
to standardize: z =
x

we did in section 2.1.
, as
If the variable we standardize has a
Normal distribution, then so does
the new variable z. This is true
because standardizing is a linear
transformation and does not
change the shape of a distribution.
This new distribution is called the
Standard Normal distribution.
The notation for the standard
Normal distribution is ______.
N ( 0 ,1)
The standard Normal distribution is a
density curve. Any question about the
proportion of observations can be
answered by finding the area under
the curve. Because all Normal
distributions are the same when we
standardize, we can find areas under
any Normal curve from a table (or
calculator).
Turn to Table A in the front of your
text.
1  . 9066  . 0934
Calculator:
2nd VARS (DISTR)
2:normalcdf(
ENTER
normalcdf(lower limit, upper limit)
normalcdf(1.32, 10000)
. 0934
. 8413  . 2843  . 5570
normalcdf(-.57, 1)
. 5570
Normal Distribution Calculations
We can answer any question about
proportions of observations in a
Normal distribution by
____________
standardizing and then using the
Standard Normal table.
Here is a recipe to do so.
1. State the problem in terms of the observed
variable x. Draw a picture of the distribution and
find
_____the
area of interest under the curve.
2. Standardize x to restate the problem in terms
of a standard Normal variable z.
3. Use the table or calculator and find the
required area under the standard Normal curve.
4. Write your conclusion in the context of the
problem.
z
240  170
 2 . 33
30
1  . 9901  . 0099
0.99% of all 14-year old boys have a
cholesterol level of more than 240 mg/dl
Calculator:
normalcdf(240,10000,  ,  )
normalcdf(240, 10000, 170, 30)
. 0098
z
150  170
  . 67
z
190  170
30
30
. 7486  . 2514  . 4972
49.7 % of 14-year old boys have blood
cholesterol between 150 & 190 mg/dl
 . 67
Calculator:
normalcdf(150,190,170,30)
. 4950
 . 67 
x  170
30
 . 67
x  149 . 9
25% of 14-year old boys have blood
cholesterol levels less than 149.9mg/dl
Calculator:
2nd VARS (DISTR)
3: invNorm(
ENTER
invNorm(area to left,  ,  )
invNorm(.25, 170, 30)
x  149 . 765
. 84 
x  170
30
x  195 . 2
. 84
80% of 14-year old boys have blood
cholesterol levels less than 195.2
invNorm(.8, 170, 30)
x  195 . 249