ap statistics: section 2.2 b. the standard normal distribution as the 68-95-99.7 rule suggests, all...
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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 = , as
we did in section 2.1.
x
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 ______.)1,0(N
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.
0934.9066.1
Calculator:2nd VARS (DISTR)2:normalcdf(ENTERnormalcdf(lower limit, upper limit)
normalcdf(1.32, 10000)
0934.
5570.2843.8413.
normalcdf(-.57, 1)
5570.
Normal Distribution Calculations
We can answer any question about proportions of observations in a
Normal distribution by ____________ and then using the
Standard Normal table. standardizing
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
_____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.
find
33.230
170240
z
0099.9901.1
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.
67.30
170150
z 67.
30
170190
z
4972.2514.7486. 49.7 % of 14-year old boys have blood cholesterol between 150 & 190 mg/dl
Calculator:
normalcdf(150,190,170,30)
4950.
67.
30
17067.
x
9.149x
25% of 14-year old boys have blood cholesterol levels less than 149.9mg/dl
Calculator:2nd VARS (DISTR)3: invNorm(ENTERinvNorm(area to left,
invNorm(.25, 170, 30)
),
765.149x
84.
30
17084.
x
2.195x
80% of 14-year old boys have blood cholesterol levels less than 195.2
invNorm(.8, 170, 30)
249.195x