AP Statistics Curriculum 2007 Normal Std

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=== Standard Normal Distrribution===
=== Standard Normal Distrribution===
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The standard normal distribution is a continuous distribution where the
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The standard normal distribution 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:
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It 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.
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* The area: -1 < z < 1 = 0.8413 - 0.1587 = 0.6826
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* The area: -2.0 < z < 2.0 = 0.9772 - 0.0228 = 0.9544 
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* The area: -3.0 < z < 3.0 = 0.9987 - 0.0013 = 0.9974
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig10.jpg|500px]]</center>
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* The Standard Normal distribution is also a special case of the [[AP_Statistics_Curriculum_2007_Normal_Prob | 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===
===Experiments===
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Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!
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Suppose we decide to test the state of 100 used batteries. To do that, we connet 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 <math>X_i</math>={measured voltage for battery i} - this is random variable 0 and assume the distribution of all <math>X_i</math> is Standard Normal, <math>X_i \sim N(0,1)</math>. Use the Normal Distribution (with mean=0 and variance=1) in the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distribution applet] to address the follwoing questions. This [[Help_pages_for_SOCR_Distributions | Disrtibutions help-page may be useful in understanding SOCR Distribution Applet]]. How many batteries, from the sample of 100, can we expect to have:
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<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
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* Absolute Voltage > 1? P(X>1) = 0.1586, thus we expect 15-16 batteries to have voltage exceeding 1.
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig1.jpg|500px]]</center>
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===Approach===
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* |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
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Models & strategies for solving the problem, data understanding & inference.  
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig2.jpg|500px]]</center>
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* Voltage < -2? P(X<-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than -2.
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* TBD
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]</center>
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* Voltage <= -2? P(X<=-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than or equal to -2.
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===Model Validation===
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]</center>
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Checking/affirming underlying assumptions.  
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* -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465) = 0.761622, thus we expect 76 batteries to have voltage in this range.
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig4.jpg|500px]]</center>
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* TBD
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===Computational Resources: Internet-based SOCR Tools===
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* TBD
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===Examples===
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Computer simulations and real observed data.  
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* TBD
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===Hands-on activities===
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Step-by-step practice problems.  
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* TBD
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<hr>
<hr>
===References===
===References===
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* TBD
 
<hr>
<hr>

Revision as of 19:57, 31 January 2008

Contents

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

Standard Normal Distrribution

The standard normal distribution 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 < z < 1 = 0.8413 - 0.1587 = 0.6826
  • The area: -2.0 < z < 2.0 = 0.9772 - 0.0228 = 0.9544
  • The area: -3.0 < z < 3.0 = 0.9987 - 0.0013 = 0.9974
  • 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 connet 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 Xi={measured voltage for battery i} - this is random variable 0 and assume the distribution of all Xi 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 follwoing questions. This Disrtibutions 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




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