AP Statistics Curriculum 2007 Chi-Square

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.

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, +∞)

Moments:
The nth raw moment for a distribution with r degrees of freedom is\[2^n \tfrac{\Gamma(n+\tfrac{1}{2}r)}{\Gamma\tfrac{1}{2}r}\]

The nth central moment is\[2^nU(-n,1-n-\tfrac{1}{2}r,-\tfrac{1}{2}r)\],

where U(a,b,x) is a confluent hypergeometric function of the second kind.

Applications

• See the Chi-square Distribution use to compute confidence intervals of variances
• Chi-Square goodness of fit
• Independence of two criteria of classification of qualitative data
• Confidence Interval estimation for a population standard deviation of a normal distribution from a sample standard deviation
• 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)	\((Obs-Exp)^2\)	\((Obs-Exp)^2/Exp\)
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.

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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• SOCR Home page: http://www.socr.ucla.edu

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