# AP Statistics Curriculum 2007 Normal Std

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=== Standard Normal Distribution=== | === Standard Normal Distribution=== | ||

The Standard Normal Distribution is a continuous distribution with the following density: | The Standard Normal Distribution is a continuous distribution with the following density: | ||

- | * Standard Normal ''density'' function <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}.</math> | + | * Standard Normal ''density'' function <math>f(x)= {e^{-x^2 \over 2} \over \sqrt{2 \pi}}.</math> |

- | * Standard Normal ''cumulative distribution'' function <math>\Phi(y)= \int_{-\infty}^{y}{{e^{-x^2} \over \sqrt{2 \pi}} dx}.</math> | + | * Standard Normal ''cumulative distribution'' function <math>\Phi(y)= \int_{-\infty}^{y}{{e^{-x^2 \over 2} \over \sqrt{2 \pi}} dx}.</math> |

Note that the following exact ''areas'' are bound between the Standard Normal Density Function and the x-axis on these symmetric intervals around the origin: | Note that the following exact ''areas'' are bound between the Standard Normal Density Function and the x-axis on these symmetric intervals around the origin: |

## Revision as of 16:11, 18 May 2009

## Contents |

## General Advance-Placement (AP) Statistics Curriculum - Standard Normal Variables and Experiments

### Standard Normal Distribution

The Standard Normal Distribution is a continuous distribution with the following density:

- Standard Normal
*density*function - Standard Normal
*cumulative distribution*function

Note that the following exact *areas* are bound between the Standard Normal Density Function and the x-axis on these symmetric intervals around the origin:

- The area: -1 < x < 1 = 0.8413 - 0.1587 = 0.6826
- The area: -2.0 < x < 2.0 = 0.9772 - 0.0228 = 0.9544
- The area: -3.0 < x < 3.0 = 0.9987 - 0.0013 = 0.9974
- Note that the inflection points (
*f*''(*x*) = 0)of the Standard Normal density function are 1.

- The Standard Normal distribution is also a special case of the more general normal distribution where the mean is set to zero and a variance to one. The Standard Normal distribution is often called the
*bell curve*because the graph of its probability density resembles a bell.

### Experiments

Suppose we decide to test the state of 100 used batteries. To do that, we connect each battery to a volt-meter by randomly attaching the positive (+) and negative (-) battery terminals to the corresponding volt-meter's connections. Electrical current always flows from + to -, i.e., the current goes in the direction of the voltage drop. Depending upon which way the battery is connected to the volt-meter we can observe positive or negative voltage recordings (voltage is just a difference, which forces current to flow from higher to the lower voltage.) Denote *X*_{i}={measured voltage for battery i} - this is random variable with mean of 0 and unitary variance. Assume the distribution of all *X*_{i} is Standard Normal, . Use the Normal Distribution (with mean=0 and variance=1) in the SOCR Distribution applet to address the following questions. This Distributions help-page may be useful in understanding SOCR Distribution Applet. How many batteries, from the sample of 100, can we expect to have?

- Absolute Voltage > 1? P(X>1) = 0.1586, thus we expect 15-16 batteries to have voltage exceeding 1.

- |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173, thus we expect 31-32 batteries to have absolute voltage exceeding 1.

- Voltage < -2? P(X<-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than -2.

- Voltage <= -2? P(X<=-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than or equal to -2.

- -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465) = 0.761622, thus we expect 76 batteries to have voltage in this range.

### References

- SOCR Home page: http://www.socr.ucla.edu

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