AP Statistics Curriculum 2007 Contingency Fit

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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Multinomial Experiments: Goodness-of-Fit ==
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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Multinomial Experiments: Chi-Square Goodness-of-Fit ==
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=== Multinomial Experiments: Goodness-of-Fit ===
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The chi-square test is used to test if a data sample comes from a population with a specific characteristics. The chi-square goodness-of-fit test is applied to binned data (data put into classes or categoris). In most situations, the data histogram or frequency histogram may be obtained and the chi-square test may be applied to these (frequency) values. The chi-square test requires a sufficient sample size in order for the chi-square approximation to be valid.
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Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!
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<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
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===Approach===
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The [http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test Kolmogorov-Smirnov] is an alternative to the Chi-square goodness-of-fit test. The chi-square goodness-of-fit test may also be applied to discrete distributions such as the binomial and the Poisson. The Kolmogorov-Smirnov test is restricted to continuous distributions.
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Models & strategies for solving the problem, data understanding & inference.  
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* TBD
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==Motivational example==
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[http://en.wikipedia.org/wiki/Mendelian_inheritance Mendel's pea experiment] relates to the transmission of hereditary characteristics from parent organisms to their offspring; it underlies much of genetics. Suppose a ''tall offspring'' is the event of interest and that the true proportion of tall peas (based on a 3:1 phenotypic ratio) is 3/4 or ''p = 0.75''.  He would like to show that Mendel's data follow this 3:1 phenotypic ratio.
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===Model Validation===
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<center>
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Checking/affirming underlying assumptions.
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{| class="wikitable" style="text-align:center; width:25%" border="1"
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|-
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|  || '''Observed''' (O) || '''Expected''' (E)
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|-
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| '''Tall''' || 787 || 798
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|-
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| '''Dwarf'''|| 277 || 266
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|}
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</center>
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* TBD
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==Calculations==
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===Computational Resources: Internet-based SOCR Tools===
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Suppose there were ''N = 1064''  data measurements with ''Observed(Tall) = 787'' and ''Observed(Dwarf) = 277''. These are the O’s (observed values). To calculate the E’s (expected values), we will take the hypothesized proportions under <math>H_o</math> and multiply them by the total sample size ''N''. Expected(Tall) = (0.75)(1064) = 798 and Expected(Dwarf) = (0.25)(1064) = 266
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* TBD
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Quickly check to see if the expected total = N = 1064.
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===Examples===
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* The hypotheses:
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Computer simulations and real observed data.  
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: <math>H_o</math>:P(tall) = 0.75 (No effect, follows a 3:1phenotypic ratio)
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:: P(dwarf) = 0.25 
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: <math>H_a</math>: P(tall)  ≠  0.75
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::P(dwarf) ≠ 0.25
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* TBD
 
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===Hands-on activities===
 
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Step-by-step practice problems.
 
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* TBD
 
<hr>
<hr>

Revision as of 01:48, 3 March 2008

Contents

General Advance-Placement (AP) Statistics Curriculum - Multinomial Experiments: Chi-Square Goodness-of-Fit

The chi-square test is used to test if a data sample comes from a population with a specific characteristics. The chi-square goodness-of-fit test is applied to binned data (data put into classes or categoris). In most situations, the data histogram or frequency histogram may be obtained and the chi-square test may be applied to these (frequency) values. The chi-square test requires a sufficient sample size in order for the chi-square approximation to be valid.

The Kolmogorov-Smirnov is an alternative to the Chi-square goodness-of-fit test. The chi-square goodness-of-fit test may also be applied to discrete distributions such as the binomial and the Poisson. The Kolmogorov-Smirnov test is restricted to continuous distributions.

Motivational example

Mendel's pea experiment relates to the transmission of hereditary characteristics from parent organisms to their offspring; it underlies much of genetics. Suppose a tall offspring is the event of interest and that the true proportion of tall peas (based on a 3:1 phenotypic ratio) is 3/4 or p = 0.75. He would like to show that Mendel's data follow this 3:1 phenotypic ratio.

Observed (O) Expected (E)
Tall 787 798
Dwarf 277 266

Calculations

Suppose there were N = 1064 data measurements with Observed(Tall) = 787 and Observed(Dwarf) = 277. These are the O’s (observed values). To calculate the E’s (expected values), we will take the hypothesized proportions under Ho and multiply them by the total sample size N. Expected(Tall) = (0.75)(1064) = 798 and Expected(Dwarf) = (0.25)(1064) = 266 Quickly check to see if the expected total = N = 1064.

  • The hypotheses:
Ho:P(tall) = 0.75 (No effect, follows a 3:1phenotypic ratio)
P(dwarf) = 0.25
Ha: P(tall) ≠ 0.75
P(dwarf) ≠ 0.25



References

  • TBD



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