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

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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. | 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. | ||

− | * The back and forth linear transformations converting between Standard and General Normal distributions are | + | * 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>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>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. | : <math>X = \mu +Z\sigma</math> converts standard scores to general normal values. | ||

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====Textbook prices==== | ====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 | + | 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 guarantee 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> | <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig12.jpg|500px]]</center> | ||

## Revision as of 17:16, 12 February 2008

## 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 always 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.

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

#### 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 guarantee 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, \(x_o=\mu +z_o\sigma=195+(-0.5243987892920383)*20=184.512024214159234.\)

#### 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 10
^{th}percentile (shortest 10% for women) in height for college women, then what is the largest height closest to hers?

### 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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