Difference between revisions of "AP Statistics Curriculum 2007 Normal Critical"

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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Nonstandard Normal Distribution & Experiments: Finding Critical Values==
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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Non-Standard Normal Distribution and Experiments: Finding Critical Values==
  
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=== Nonstandard Normal Distribution & Experiments: Finding Scores (Critical Values)===
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=== Non-Standard Normal Distribution and Experiments: Finding Scores (Critical Values)===
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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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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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* 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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* The back and forth linear transformations converting between Standard and General Normal Distributions are always useful in such analyses (Let ''X'' denotes General (<math>X\sim N(\mu,\sigma^2)</math>) and ''Z'' denotes Standard (<math>Z\sim N(0,1)</math>) Normal random variables):
 
: <math>Z = {X-\mu \over \sigma}</math> converts general normal scores to standard (Z) values.
 
: <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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: <math>X = \mu +Z\sigma \ </math> converts standard scores to general normal values.
  
 
===Examples===
 
===Examples===
 
This [[Help_pages_for_SOCR_Distributions | Distributions help-page may be useful in understanding SOCR Distribution Applet]].  
 
This [[Help_pages_for_SOCR_Distributions | Distributions help-page may be useful in understanding SOCR Distribution Applet]].  
  
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====Textbook prices====
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====[[SOCR_EduMaterials_Activities_Normal_Probability_examples | A large number of Normal distribution examples using SOCR tools is available here]]====
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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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====Textbook Prices====
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Suppose the amount of money that 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 she spent on books this semester, what is the maximum dollar amount that would guarantee she spends only as little as 30% of the population? (<math>P(X<184.512)=0.3</math>)
 
<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig12.jpg|500px]]</center>
 
<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig12.jpg|500px]]</center>
  
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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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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 \(z_o=-0.5243987892920383\), then \(P(-z_o<Z<z_o)=0.4\) and \(P(Z<z_o)=0.3\). Thus, \(x_o=\mu +z_o\sigma=195+(-0.5243987892920383)*20=184.512024214159234\).
 
<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig13.jpg|500px]]</center>
 
<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig13.jpg|500px]]</center>
  
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<center>
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====Human Weights and Heights====
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{| class="wikitable" style="text-align:center; width:75%" border="1"
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Human [http://en.wikipedia.org/wiki/Human_weight weights] and [http://en.wikipedia.org/wiki/Human_height heights] are known to be approximately Normally distributed. Look at the [[SOCR_Data_Dinov_020108_HeightsWeights |SOCR Weighs and Heights Dataset]] and use the [[SOCR_EduMaterials_Activities_Histogram_Graphs | SOCR Charts]] to validate these statements based on this sample dataset.
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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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* Suppose the heights of college women are approximately Normally distributed with a mean of 65 inches and a standard deviation of 2 inches. If a randomly chosen college woman is at the 10<sup>th</sup> percentile (shortest 10% for women) in height for college women, then what is the largest height closest to hers (i.e., what is her estimated height)?
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|}</center>
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<center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig14.jpg|500px]]</center>
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===[[EBook_Problems_Normal_Critical|Problems]]===
  
 
<hr>
 
<hr>
  
 
===References===
 
===References===
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* [[SOCR_EduMaterials_Activities_Histogram_Graphs | Histogram plots]]  
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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_BoxPlot | Box-and-whisker Plots]]
 
* [[SOCR_EduMaterials_Activities_DotChart |Dotplot]]  
 
* [[SOCR_EduMaterials_Activities_DotChart |Dotplot]]  
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* [[SOCR_EduMaterials_Activities_QQChart |Quantile-Quantile probability plot]]  
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* [[SOCR_EduMaterials_Activities_QQChart |Quantile-Quantile Probability Plot]]  
 
* [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR High-Precision Normal Distribution Calculator]
 
* [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR High-Precision Normal Distribution Calculator]
  
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* SOCR Home page: http://www.socr.ucla.edu
 
* SOCR Home page: http://www.socr.ucla.edu
  
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Normal_Critical}}
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"{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Normal_Critical}}

Latest revision as of 13:04, 3 March 2020

General Advance-Placement (AP) Statistics Curriculum - Non-Standard Normal Distribution and Experiments: Finding Critical Values

Non-Standard Normal Distribution and 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 always useful in such analyses (Let X denotes General (\(X\sim N(\mu,\sigma^2)\)) and Z denotes Standard (\(Z\sim N(0,1)\)) Normal random variables):

\[Z = {X-\mu \over \sigma}\] converts general normal scores to standard (Z) values. \[X = \mu +Z\sigma \ \] converts standard scores to general normal values.

Examples

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

A large number of Normal distribution examples using SOCR tools is available here

Textbook Prices

Suppose the amount of money that 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 she spent on books this semester, what is the maximum dollar amount that would guarantee she spends only as little as 30% of the population? (\(P(X<184.512)=0.3\))

SOCR EBook Dinov RV Normal 013108 Fig12.jpg

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

SOCR EBook Dinov RV Normal 013108 Fig13.jpg

Human Weights and Heights

Human weights and heights are known to be approximately Normally distributed. Look at the SOCR Weighs and Heights Dataset and use the SOCR Charts to validate these statements based on this sample dataset.

  • Suppose the heights of college women are approximately Normally distributed with a mean of 65 inches and a standard deviation of 2 inches. If a randomly chosen college woman is at the 10th percentile (shortest 10% for women) in height for college women, then what is the largest height closest to hers (i.e., what is her estimated height)?
SOCR EBook Dinov RV Normal 013108 Fig14.jpg

Problems

References

"-----

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