AP Statistics Curriculum 2007 Chi-Square - Revision history

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Dinov: /* Chi-Square Distribution */

2014-10-26T15:12:27Z

‎Chi-Square Distribution

IvoDinov: /* Applications */

2012-02-27T16:57:05Z

‎Applications

JayZzz: /* Chi-Square Distribution */

2011-07-18T22:46:22Z

‎Chi-Square Distribution

JayZzz at 03:59, 3 July 2011

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JayZzz at 21:38, 2 July 2011

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JayZzz: Created page with '== General Advance-Placement (AP) Statistics Curriculum - Chi-Square Distribution== === Chi-Square Distribution === The Chi-Square distributio…'

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Created page with '== General Advance-Placement (AP) Statistics Curriculum - Chi-Square Distribution== === Chi-Square Distribution === The Chi-Square distributio…'

New page

==[[AP_Statistics_Curriculum_2007 | 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''': 
<math>\frac{1}{2^{k/2}\Gamma(k/2)}\; x^{k/2-1} e^{-x/2}\,</math>

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

'''Mean''': 
<math>\approx k\bigg(1-\frac{2}{9k}\bigg)^3</math>

'''Median''': 
<math>\approx k\bigg(1-\frac{2}{9k}\bigg)^3</math>

'''Mode''': 
max{ ''k'' − 2, 0 }

'''Variance''': 
2''k''

'''Support''': 
''x'' ∈ [0, +∞)

'''1st Moment''': 
k

'''2nd Moment''': 
2k

===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 Independence] of two criteria of classification of qualitative data

<math>\cdot</math> [http://en.wikipedia.org/wiki/Confidence_interval Confidence Interval] estimation for a population standard deviation of a normal distribution from a sample standard deviation

<math>\cdot</math> [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===
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.

{| class="wikitable"
|-
! Birthday Month
! Observed
! Expected
! Residual (Obs-Exp)
! <math>(Obs-Exp)^2</math>
! <math>(Obs-Exp)^2/Exp</math>
|-
| '''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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 '''Support''': <br>
-''x'' ∈ [0, +∞)http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Chi-Square&action=edit
+''x'' ∈ [0, +∞)
 '''Moments''': <br>

@@ Line 35: / Line 35: @@
 ===Applications===
-<math>\cdot</math> [http://en.wikipedia.org/wiki/Goodness_of_fit Chi-Square goodness of fit]
+* 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]
-<math>\cdot</math> [http://en.wikipedia.org/wiki/Goodness_of_fit Independence] of two criteria of classification of qualitative data
+* [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
-<math>\cdot</math> [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===

@@ Line 2: / Line 2: @@
 === 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].
+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>
@@ Line 32: / Line 32: @@
 <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===

@@ Line 23: / Line 23: @@
 '''Support''': <br>
-''x'' ∈ [0, +∞)
+''x'' ∈ [0, +∞)http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Chi-Square&action=edit
-====Raw Moments====
+'''Moments''': <br>
-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''.
+The nth raw moment for a distribution with r degrees of freedom is:
-====Centralized Moments====
+The nth central moment is:
-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''.
+<math>2^nU(-n,1-n-\tfrac{1}{2}r,-\tfrac{1}{2}r)</math>,
-====Standardized Moments====
+where U(a,b,x) is a [http://en.wikipedia.org/wiki/Confluent_hypergeometric_function confluent hypergeometric function] of the second kind.
 ===Applications===
 <math>\cdot</math> [http://en.wikipedia.org/wiki/Goodness_of_fit Chi-Square goodness of fit]

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