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

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Page 1: 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

AP Statistics: Section 2.2 B

Page 2: 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

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.

Page 3: 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

Changing to these units requires us

to standardize: z = , as

we did in section 2.1.

x

Page 4: 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

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.

Page 5: 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

The notation for the standard Normal distribution is ______.)1,0(N

Page 6: 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

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).

Page 7: 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

Turn to Table A in the front of your text.

Page 8: 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

Page 9: 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

Page 10: 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

0934.9066.1

Page 11: 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

Calculator:2nd VARS (DISTR)2:normalcdf(ENTERnormalcdf(lower limit, upper limit)

normalcdf(1.32, 10000)

0934.

Page 12: 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

Page 13: 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

Page 14: 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

5570.2843.8413.

Page 15: 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

normalcdf(-.57, 1)

5570.

Page 16: 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

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

Page 17: 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

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

Page 18: 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

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

Page 19: 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

Calculator:

normalcdf(240,10000,

normalcdf(240, 10000, 170, 30)

),

0098.

Page 20: 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

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

Page 21: 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

Calculator:

normalcdf(150,190,170,30)

4950.

Page 22: 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

Page 23: 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

Page 24: 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

67.

30

17067.

x

9.149x

25% of 14-year old boys have blood cholesterol levels less than 149.9mg/dl

Page 25: 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

Calculator:2nd VARS (DISTR)3: invNorm(ENTERinvNorm(area to left,

invNorm(.25, 170, 30)

),

765.149x

Page 26: 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

Page 27: 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

Page 28: 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

84.

30

17084.

x

2.195x

80% of 14-year old boys have blood cholesterol levels less than 195.2

Page 29: 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

invNorm(.8, 170, 30)

249.195x

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