AP Statistics Curriculum 2007 Normal Std

Revision as of 18:03, 1 March 2008 by PriscillaChui (talk | contribs) (Standard Normal Distribution)
Jump to: navigation, search

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

Standard Normal Distribution

The Standard Normal Cistribution is a continuous distribution where the following exact areas are bound between the Standard Normal Density Function and the x-axis on the 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
Error creating thumbnail: File missing
  • Standard Normal density function \(f(x)= {e^{-x^2} \over \sqrt{2 \pi}}.\)
  • 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.


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 0 and assume the distribution of all \(X_i\) is Standard Normal, \(X_i \sim N(0,1)\). 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.
SOCR EBook Dinov RV Normal 013108 Fig1.jpg
  • |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
SOCR EBook Dinov RV Normal 013108 Fig2.jpg
  • Voltage < -2? P(X<-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than -2.
SOCR EBook Dinov RV Normal 013108 Fig3.jpg
  • Voltage <= -2? P(X<=-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than or equal to -2.
SOCR EBook Dinov RV Normal 013108 Fig3.jpg
  • -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465) = 0.761622, thus we expect 76 batteries to have voltage in this range.
SOCR EBook Dinov RV Normal 013108 Fig4.jpg


Translate this page:

Uk flag.gif

De flag.gif

Es flag.gif

Fr flag.gif

It flag.gif

Pt flag.gif

Jp flag.gif

Bg flag.gif

الامارات العربية المتحدة
Ae flag.gif

Fi flag.gif

इस भाषा में
In flag.gif

No flag.png

Kr flag.gif

Cn flag.gif

Cn flag.gif

Ru flag.gif

Nl flag.gif

Gr flag.gif

Hr flag.gif

Česká republika
Cz flag.gif

Dk flag.gif

Pl flag.png

Ro flag.png

Se flag.gif