# AP Statistics Curriculum 2007 Contingency Fit

(Difference between revisions)
 Revision as of 19:10, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 01:50, 3 March 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 1: Line 1: - ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Multinomial Experiments: Goodness-of-Fit == + ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Multinomial Experiments: Chi-Square Goodness-of-Fit == - === Multinomial Experiments: 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. - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
+ - ===Approach=== + 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. - Models & strategies for solving the problem, data understanding & inference. + - * TBD + ==Motivational example== + [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. - ===Model Validation=== +
- Checking/affirming underlying assumptions. + {| class="wikitable" style="text-align:center; width:25%" border="1" + |- + |  || '''Observed''' (O) || '''Expected''' (E) + |- + | '''Tall''' || 787 || 798 + |- + | '''Dwarf'''|| 277 || 266 + |} +
- * TBD + ==Calculations== - ===Computational Resources: Internet-based SOCR Tools=== + 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 $H_o$ and multiply them by the total sample size ''N''. Expected(Tall) = (0.75)(1064) = 798 and Expected(Dwarf) = (0.25)(1064) = 266 - * TBD + Quickly check to see if the expected total = N = 1064. - ===Examples=== + * The hypotheses: - Computer simulations and real observed data. + : $H_o$:P(tall) = 0.75 (No effect, follows a 3:1phenotypic ratio) + :: P(dwarf) = 0.25 + : $H_a$: P(tall)  ≠  0.75 + ::P(dwarf) ≠ 0.25 - * TBD - - ===Hands-on activities=== - Step-by-step practice problems. - - * TBD

## 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

• TBD