AP Statistics Curriculum 2007 Chi-Square
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=== 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 | + | 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 [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Std normal distribution] from a sample standard deviation. The Chi-Square distribution is a special case of the [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Gamma Gamma distribution]. |
'''PDF''': <br> | '''PDF''': <br> | ||
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'''Support''': <br> | '''Support''': <br> | ||
- | ''x'' ∈ [0, +∞) | + | ''x'' ∈ [0, +∞)http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Chi-Square&action=edit |
- | ''' | + | '''Moments''': <br> |
- | + | The nth raw moment for a distribution with r degrees of freedom is: | |
+ | <math>2^n \tfrac{\Gamma(n+\tfrac{1}{2}r)}{\Gamma\tfrac{1}{2}r}</math> | ||
- | + | The nth central moment is: | |
- | + | <math>2^nU(-n,1-n-\tfrac{1}{2}r,-\tfrac{1}{2}r)</math>, | |
+ | |||
+ | where U(a,b,x) is a [http://en.wikipedia.org/wiki/Confluent_hypergeometric_function confluent hypergeometric function] of the second kind. | ||
===Applications=== | ===Applications=== | ||
- | + | * See the [[AP_Statistics_Curriculum_2007_Estim_Var| Chi-square Distribution use to compute confidence intervals of variances]] | |
- | + | * [http://en.wikipedia.org/wiki/Goodness_of_fit Chi-Square goodness of fit] | |
- | + | * [http://en.wikipedia.org/wiki/Goodness_of_fit Independence] of two criteria of classification of qualitative data | |
- | + | * [http://en.wikipedia.org/wiki/Confidence_interval Confidence Interval] estimation for a population standard deviation of a normal distribution from a sample standard deviation | |
- | + | * [http://en.wikipedia.org/wiki/ANOVA 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=== | ===Example=== |
Current revision as of 16:51, 27 February 2012
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.
PDF:
CDF:
Mean:
Median:
Mode:
max{ k − 2, 0 }
Variance:
2k
Support:
x ∈ [0, +∞)http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Chi-Square&action=edit
Moments:
The nth raw moment for a distribution with r degrees of freedom is:
The nth central moment is: ,
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)
- SOCR Home page: http://www.socr.ucla.edu
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