# AP Statistics Curriculum 2007 Contingency Fit

(Difference between revisions)
 Revision as of 01:50, 3 March 2008 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 02:10, 3 March 2008 (view source)IvoDinov (Talk | contribs) (→Calculations)Newer edit → Line 21: Line 21: ==Calculations== ==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 $H_o$ and multiply them by the total sample size ''N''. Expected(Tall) = (0.75)(1064) = 798 and Expected(Dwarf) = (0.25)(1064) = 266 + 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. Quickly check to see if the expected total = N = 1064. - Quickly check to see if the expected total = N = 1064. + * The hypotheses: * The hypotheses: Line 29: Line 28: : $H_a$: P(tall)  ≠  0.75 : $H_a$: P(tall)  ≠  0.75 ::P(dwarf) ≠ 0.25 ::P(dwarf) ≠ 0.25 + + * Test statistics: + :$\chi_o^2 = \sum_{all-categories}{(O-E)^2/E} \sim \chi_{df=number\_of\_categories - 1}^2$ + + * P-values and critical values for the [http://socr.stat.ucla.edu/htmls/SOCR_Distributions.html Chi-Square distribution may be easily computed using SOCR Distributions]. + + * Results: + For the Mendel's pea experiment, we can compute the Chi-square test statistics to be: + : $\chi_o^2 = {(787-798)^2 \over 798} + {(277-266)@ \over 266} = 0.152+0.455=0.607$. + : p-value=$P(Chi^2 > \chi_o^2)=0.436$ + + * SOCR Chi-square Caluclations + +
[[Image:SOCR_EBook_Dinov_ChiSquare_030108_Fig1.jpg|500px]]

## 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
• Test statistics:
$\chi_o^2 = \sum_{all-categories}{(O-E)^2/E} \sim \chi_{df=number\_of\_categories - 1}^2$
• Results:

For the Mendel's pea experiment, we can compute the Chi-square test statistics to be:

Failed to parse (lexing error): \chi_o^2 = {(787-798)^2 \over 798} + {(277-266)@ \over 266} = 0.152+0.455=0.607

.

p-value=$P(Chi^2 > \chi_o^2)=0.436$
• SOCR Chi-square Caluclations

• TBD