# AP Statistics Curriculum 2007 Normal Critical

## 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.

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