ap statistics: section 2.2 b. the standard normal distribution as the 68-95-99.7 rule suggests, all...

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