AP Statistics Curriculum 2007 Chi-Square

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''x'' ∈ [0, +∞)
''x'' ∈ [0, +∞)
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'''1st Moment''': <br>
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====Raw Moments====
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k
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The ''k''<sup>th</sup> '''Raw Moment''' for a discrete random variable ''X'' is defined by <math>E[X^k]=\sum_x{x^kP(X=x)}.</math> The ''k''<sup>th</sup> '''Raw Moment''' for a continuously-values random variable ''Y'' is analogously defined by <math>E[Y^k]=\int{y^kP(y)dy},</math> where the integral is over the domain of ''Y'' and ''P(y)'' is the probability density function of ''Y''.
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'''2nd Moment''': <br>
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====Centralized Moments====
-
2k
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The ''k''<sup>th</sup> '''Centralized Moment''' for a discrete random variable ''X'' is defined by <math>E_c[X^k]=\sum_x{(x-\mu)^kP(X=x)},</math> where <math>\mu</math> is the expected value of ''X''. The ''k''<sup>th</sup> '''Centralized Moment''' for a continuously-values random variable ''Y'' is analogously defined by <math>E_c[Y^k]=\int{(y-\mu)^kP(y)dy},</math> where <math>\mu</math> is the expected value of ''Y'', the integral is over the domain of ''Y'' and ''P(y)'' is the probability density function of ''Y''.
 +
====Standardized Moments====
 +
The ''k''<sup>th</sup> '''Standardized Moment''' for a discrete random variable ''X'' is defined by
 +
 +
: <math>E_s[X^k]={\sum_x{(x-\mu)^kP(X=x)} \over {(\sum_{x} (x-\mu)^2P(X=x))^{k/2}}}.</math>
 +
 +
The ''k''<sup>th</sup> '''Standardized Moment''' for a continuously-values random variable ''Y'' is analogously defined by
 +
 +
:<math>E_s[Y^k]={\int{(y-\mu)^kP(y)dy} \over \sigma^k},</math> where the integral is over the domain of ''Y'' and ''P(y)'' is the probability density function of ''Y''
===Applications===
===Applications===
<math>\cdot</math> [http://en.wikipedia.org/wiki/Goodness_of_fit Chi-Square goodness of fit]
<math>\cdot</math> [http://en.wikipedia.org/wiki/Goodness_of_fit Chi-Square goodness of fit]

Revision as of 21:32, 2 July 2011

Contents

General Advance-Placement (AP) Statistics Curriculum - Chi-Square Distribution

Chi-Square Distribution

The Chi-Square distribution is used in the chi-square tests for goodness of fit of an observed distribution to a theoretical one and the independence of two criteria of classification of qualitative data. It is also used in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. The Chi-Square distribution is a special case of the Gamma distribution [link to gamma].

PDF:
\frac{1}{2^{k/2}\Gamma(k/2)}\; x^{k/2-1} e^{-x/2}\,

CDF:
\frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}\!

Mean:
\approx k\bigg(1-\frac{2}{9k}\bigg)^3

Median:
\approx k\bigg(1-\frac{2}{9k}\bigg)^3

Mode:
max{ k − 2, 0 }

Variance:
2k

Support:
x ∈ [0, +∞)

Raw Moments

The kth Raw Moment for a discrete random variable X is defined by
E[Xk] = xkP(X = x).
x
The kth Raw Moment for a continuously-values random variable Y is analogously defined by E[Y^k]=\int{y^kP(y)dy}, where the integral is over the domain of Y and P(y) is the probability density function of Y.

Centralized Moments

The kth Centralized Moment for a discrete random variable X is defined by
Ec[Xk] = (x − μ)kP(X = x),
x
where μ is the expected value of X. The kth Centralized Moment for a continuously-values random variable Y is analogously defined by E_c[Y^k]=\int{(y-\mu)^kP(y)dy}, where μ is the expected value of Y, the integral is over the domain of Y and P(y) is the probability density function of Y.

Standardized Moments

The kth Standardized Moment for a discrete random variable X is defined by

E_s[X^k]={\sum_x{(x-\mu)^kP(X=x)} \over {(\sum_{x} (x-\mu)^2P(X=x))^{k/2}}}.

The kth Standardized Moment for a continuously-values random variable Y is analogously defined by

E_s[Y^k]={\int{(y-\mu)^kP(y)dy} \over \sigma^k}, where the integral is over the domain of Y and P(y) is the probability density function of Y

Applications

\cdot Chi-Square goodness of fit

\cdot Independence of two criteria of classification of qualitative data

\cdot Confidence Interval estimation for a population standard deviation of a normal distribution from a sample standard deviation

\cdot ANOVA: The F distribution is distribution of two independent chi-square random variables, divided by their respective degrees of freedom [link to Fisher’s F, ANOVA]

Example

Chi Square Test for Goodness of Fit: There are 60 people in a statistics class, and we have data on the month of their birth. Our null hypothesis is that the number of students with a particular birth month should be divided equally among the total 60. We can use a chi square test with 12-1=11 degrees of freedom to compare the observed data against our null hypothesis.

Birthday Month Observed Expected Residual (Obs-Exp) (ObsExp)2 (ObsExp)2 / Exp
Jan 3 5 -2 4 0.8
Feb 4 5 -1 1 0.2
Mar 8 5 3 9 1.8
April 4 5 -1 1 0.2
May 2 5 -3 9 1.8
June 3 5 -2 4 0.8
July 6 5 1 1 0.2
Aug 6 5 1 1 0.2
Sept 4 5 -1 1 0.2
Oct 3 5 -2 4 0.8
Nov 2 5 -3 9 1.8
Dec 5 5 0 0 0
Total = 8.8

Our Chi Square value is 8.8. Using the SOCR Chi-Square Distribution Calculator, at 11 degrees of freedom, a chi square value of 8.8 gives us a p-value of 0.36. We do not reject our null hypothesis. The observed data do not show evidence of a non-uniform distribution of birth months.

file:Chi-Square.png

SOCR Links

http://www.distributome.org/ -> SOCR -> Distributions -> Distributome

http://www.distributome.org/ -> SOCR -> Distributions -> Chi-Square Distribution

http://www.distributome.org/ -> SOCR -> Functors -> Chi-Square Distribution

http://www.distributome.org/ -> SOCR -> Analyses -> Chi-Square Test Contingency Table

http://www.distributome.org/ -> SOCR -> Analyses -> Chi-Square Model Goodness-of-Fit Test

http://www.distributome.org/ -> SOCR -> Modeler -> ChiSquareFit_Modeler

SOCR Chi-Square Distribution Calculator (http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html)




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