Difference between revisions of "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)=== | ||
| − | + | 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): |
| − | + | : <math>Z = {X-\mu \over \sigma}</math> converts general normal scores to standard (Z) values. | |
| + | : <math>X = \mu +Z\sigma</math> converts standard scores to general normal values. | ||
| − | + | ===Examples=== | |
| + | This [[Help_pages_for_SOCR_Distributions | 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? (<math>P(X<184.512)=0.3</math>) | |
| + | <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig12.jpg|500px]]</center> | ||
| − | * | + | 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> |
| + | <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig13.jpg|500px]]</center> | ||
| − | === | + | <center> |
| − | + | {| class="wikitable" style="text-align:center; width:75%" border="1" | |
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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 | ||
| + | |}</center> | ||
| − | + | <hr> | |
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===References=== | ===References=== | ||
| − | * | + | * [[SOCR_EduMaterials_Activities_Histogram_Graphs | Histogram plots]] |
| + | * [[SOCR_EduMaterials_Activities_BoxPlot | Box-and-whisker plots]] | ||
| + | * [[SOCR_EduMaterials_Activities_DotChart |Dotplot]] | ||
| + | * [[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] | ||
<hr> | <hr> | ||
Revision as of 20:52, 31 January 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. \[X = \mu +Z\sigma\] 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 \(z_o=-0.5243987892920383\), then \(P(-z_o<Z<z_o)=0.4\) and P(Z<z_o)=0.3. Thus, \(x_o=\mu +z_o\sigma=195+(-0.5243987892920383)*20=184.512024214159234.\)
| 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
- Histogram plots
- Box-and-whisker plots
- Dotplot
- Quantile-Quantile probability plot
- SOCR High-Precision Normal Distribution Calculator
- SOCR Home page: http://www.socr.ucla.edu
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