AP Statistics Curriculum 2007 Normal Critical
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, \(x_o=\mu +z_o\sigma=195+(-0.5243987892920383)*20=184.512024214159234.\)
Human Weights and Heights
Human weights and 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 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?
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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