# AP Statistics Curriculum 2007 Normal Critical

(Difference between revisions)
 Revision as of 18:51, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 01:46, 1 February 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 2: Line 2: === Nonstandard Normal Distribution & Experiments: Finding Scores (Critical Values)=== === Nonstandard Normal Distribution & Experiments: Finding Scores (Critical Values)=== - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + 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. -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
+ - ===Approach=== + * 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): - Models & strategies for solving the problem, data understanding & inference. + : $Z = {X-\mu \over \sigma}$ converts general normal scores to standard (Z) values. + : $X = \mu +Z\sigma$ converts standard scores to general normal values. - * TBD + ===Examples=== + This [[Help_pages_for_SOCR_Distributions | Distributions help-page may be useful in understanding SOCR Distribution Applet]]. - ===Model Validation=== + ====Textbook prices==== - Checking/affirming underlying assumptions. + 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) + [[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig12.jpg|500px]] - * TBD + 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 [itex]z_o=-0.5243987892920383$, then [itex]P(-z_o[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig13.jpg|500px]] - ===Computational Resources: Internet-based SOCR Tools=== +
- * TBD + {| class="wikitable" style="text-align:center; width:75%" border="1" + |- + | Height (in.)  || 61.0 || 62.5 || 63.0 || 64.0 || 64.5 || 65.0 || 66.5 || 67.0 || 68.0 || 68.5 || 70.5 + |}
- ===Examples=== +
- Computer simulations and real observed data. + - * TBD - - ===Hands-on activities=== - Step-by-step practice problems. - - * TBD - -
===References=== ===References=== - * TBD + * [[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]

## 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

#### 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 62.5 63 64 64.5 65 66.5 67 68 68.5 70.5