AP Statistics Curriculum 2007 Normal Prob

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Revision as of 20:19, 31 January 2008

General Advance-Placement (AP) Statistics Curriculum - Nonstandard Normal Distribution & Experiments: Finding Probabilities

General Normal Distribution

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

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.

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 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.

|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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@@ Line 1: / Line 1: @@
 ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Nonstandard Normal Distribution & Experiments: Finding Probabilities==
-=== Nonstandard Normal Distribution & Experiments: Finding Probabilities===
+=== General Normal Distribution===
-Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!
+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:
-<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
+* 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
+<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig0.jpg|500px]]</center>
-===Approach===
+* Standard Normal density function <math>f(x)= {e^{-x^2} \over \sqrt{2 \pi}}.</math>
-Models & strategies for solving the problem, data understanding & inference.
-* TBD
+* 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.
-===Model Validation===
+===Experiments===
-Checking/affirming underlying assumptions.
+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 <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 following questions. This [[Help_pages_for_SOCR_Distributions | 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.
-* TBD
+<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig1.jpg|500px]]</center>
+* |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
-===Computational Resources: Internet-based SOCR Tools===
+<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig2.jpg|500px]]</center>
-* TBD
+* Voltage < -2? P(X<-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than -2.
+<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]</center>
-===Examples===
+* Voltage <= -2? P(X<=-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than or equal to -2.
-Computer simulations and real observed data.
+<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]</center>
+* -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465) = 0.761622, thus we expect 76 batteries to have voltage in this range.
-* TBD
+<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig4.jpg|500px]]</center>
-===Hands-on activities===
-Step-by-step practice problems.
-* TBD
 <hr>
 ===References===
-* TBD
 <hr>

AP Statistics Curriculum 2007 Normal Prob

From Socr

Revision as of 20:19, 31 January 2008

Contents

General Advance-Placement (AP) Statistics Curriculum - Nonstandard Normal Distribution & Experiments: Finding Probabilities

General Normal Distribution

Experiments

References

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