AP Statistics Curriculum 2007 Normal Critical

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=== Nonstandard Normal Distribution & Experiments: Finding Scores (Critical Values)===
=== Nonstandard Normal Distribution & Experiments: Finding Scores (Critical Values)===
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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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In addition to being able to [[AP_Statistics_Curriculum_2007_Normal_Prob | compute probability (p) values]], we often need to estimate the critical values of the Normal distribution for a given p-value.
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<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
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===Approach===
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* The back and forth linear transformations converting between Standard and General Normal distributions are alwasy useful in such analyses (Let ''X'' denotes General (<math>X\sim N(\mu,\sigma^2)</math>) and ''Z'' denotes Standard (<math>X\sim N(0,1)</math>) Normal random variables):
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Models & strategies for solving the problem, data understanding & inference.  
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: <math>Z = {X-\mu \over \sigma}</math> converts general normal scores to standard (Z) values.
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: <math>X = \mu +Z\sigma</math> converts standard scores to general normal values.
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* TBD
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===Examples===
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This [[Help_pages_for_SOCR_Distributions | Distributions help-page may be useful in understanding SOCR Distribution Applet]].
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===Model Validation===
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====Textbook prices====
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Checking/affirming underlying assumptions.  
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Suppose the amount of money college students spend each semester on textbooks is normally distributed with a mean of $195 and a standard deviation of $20. If we ask a random college students from this population how much he spent on books this semester, what is the maximum dollar amount that would guagantee she spends only as much as 30% of the population? (<math>P(X<184.512)=0.3</math>)
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig12.jpg|500px]]</center>
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* TBD
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You can also do this problem exactly using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR high-precision Nornal Distribution Calculator]. If <math>z_o=-0.5243987892920383</math>, then <math>P(-z_o<Z<z_o)=0.4</math> and P(Z<z_o)=0.3. Thus, <math>x_o=\mu +z_o\sigma=195+(-0.5243987892920383)*20=184.512024214159234.</math>
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig13.jpg|500px]]</center>
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===Computational Resources: Internet-based SOCR Tools===
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<center>
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* TBD
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{| class="wikitable" style="text-align:center; width:75%" border="1"
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|-
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| Height (in.)  || 61.0 || 62.5 || 63.0 || 64.0 || 64.5 || 65.0 || 66.5 || 67.0 || 68.0 || 68.5 || 70.5
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|}</center>
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===Examples===
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<hr>
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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>
 
===References===
===References===
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* TBD
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* [[SOCR_EduMaterials_Activities_Histogram_Graphs | Histogram plots]]
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* [[SOCR_EduMaterials_Activities_BoxPlot | Box-and-whisker plots]]
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* [[SOCR_EduMaterials_Activities_DotChart |Dotplot]]
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* [[SOCR_EduMaterials_Activities_QQChart |Quantile-Quantile probability plot]]
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* [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR High-Precision Normal Distribution Calculator]
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Revision as of 01:44, 1 February 2008

Contents

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

Nonstandard Normal Distribution & Experiments: Finding Scores (Critical Values)

In addition to being able to compute probability (p) values, we often need to estimate the critical values of the Normal distribution for a given p-value.

  • The back and forth linear transformations converting between Standard and General Normal distributions are alwasy useful in such analyses (Let X denotes General (X\sim N(\mu,\sigma^2)) and Z denotes Standard (X\sim N(0,1)) Normal random variables):
Z = {X-\mu \over \sigma} converts general normal scores to standard (Z) values.
X = μ + Zσ converts standard scores to general normal values.

Examples

This Distributions help-page may be useful in understanding SOCR Distribution Applet.

Textbook prices

Suppose the amount of money college students spend each semester on textbooks is normally distributed with a mean of $195 and a standard deviation of $20. If we ask a random college students from this population how much he spent on books this semester, what is the maximum dollar amount that would guagantee she spends only as much as 30% of the population? (P(X < 184.512) = 0.3)

You can also do this problem exactly using the SOCR high-precision Nornal Distribution Calculator. If zo = − 0.5243987892920383, then P( − zo < Z < zo) = 0.4 and P(Z<z_o)=0.3. Thus, xo = μ + zoσ = 195 + ( − 0.5243987892920383) * 20 = 184.512024214159234.

Height (in.) 61.0 62.5 63.0 64.0 64.5 65.0 66.5 67.0 68.0 68.5 70.5

References




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