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**AP Statistics: Section 2.2 B**

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**The Standard Normal Distribution As the 68-95-99**

The Standard Normal Distribution As the 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.

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**Changing to these units requires us to standardize: z = , as we did in section 2.1.**

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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.

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**The notation for the standard Normal distribution is ______.**

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**The standard Normal distribution is a density curve**

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).

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**Turn to Table A in the front of your text.**

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Calculator: 2nd VARS (DISTR) 2:normalcdf( ENTER normalcdf(lower limit, upper limit) normalcdf(1.32, 10000)

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normalcdf(-.57, 1)

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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

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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 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

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**0.99% of all 14-year old boys have a cholesterol level of more than 240 mg/dl**

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**Calculator: normalcdf(240,10000, normalcdf(240, 10000, 170, 30)**

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**49.7 % of 14-year old boys have blood cholesterol between 150 & 190 mg/dl**

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**Calculator: normalcdf(150,190,170,30)**

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**25% of 14-year old boys have blood cholesterol levels less than 149**

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

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**Calculator: 2nd VARS (DISTR) 3: invNorm( ENTER invNorm(area to left, invNorm(.25, 170, 30)**

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

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invNorm(.8, 170, 30)

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C HAPTER 2: T HE N ORMAL D ISTRIBUTIONS. D ENSITY C URVES 2 A density curve describes the overall pattern of a distribution. Has an area of exactly 1.

C HAPTER 2: T HE N ORMAL D ISTRIBUTIONS. D ENSITY C URVES 2 A density curve describes the overall pattern of a distribution. Has an area of exactly 1.

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