AP Statistics Curriculum 2007 Chi-Square
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''x'' ∈ [0, +∞) | ''x'' ∈ [0, +∞) | ||
- | ''' | + | ====Raw Moments==== |
- | k | + | 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''. |
- | ''' | + | ====Centralized Moments==== |
- | + | 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 21:30, 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:
CDF:
Mean:
Median:
Mode:
max{ k − 2, 0 }
Variance:
2k
Support:
x ∈ [0, +∞)
Raw Moments
The k^{th} Raw Moment for a discrete random variable X is defined byE[X^{k}] = | ∑ | x^{k}P(X = x). |
x |
Centralized Moments
The k^{th} Centralized Moment for a discrete random variable X is defined byE_{c}[X^{k}] = | ∑ | (x − μ)^{k}P(X = x), |
x |
Standardized Moments
The k^{th} Standardized Moment for a discrete random variable X is defined by
The k^{th} Standardized Moment for a continuously-values random variable Y is analogously defined by
- where the integral is over the domain of Y and P(y) is the probability density function of Y
Applications
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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