# Difference between revisions of "Eleisha's Segment 32: Contingency Tables: A First Look"

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To Calculate:

1. 20 our of 100 U.S. Senators are women, yet when the Senate formed an intramural baseball team of 9 people only 1 woman was chosen for the team. What is the probability of this occurring by chance? What is the p-value with which the null hypothesis "there is no discrimination against women Senators" can be rejected?

$\displaystyle \text{P(Prob this occurs)} = \frac{\text{Ways to choose one female and eight males}}{\text{Total ways to choose}} = \frac{ {20 \choose 1} {80 \choose 8}}{{100 \choose 9}} \approx 0.30477.$

The null hypothesis is that there is no discrimination between men and women. The p value in this case is the probability of having no women or one woman under the null. Therefore the p-value can be calculated as:

$\displaystyle \text{p-value} = \frac{ {20 \choose 1} {80 \choose 8}}{{100 \choose 9}} + \frac{ {20 \choose 0} {80 \choose 9}}{{100 \choose 9}} \approx 0.42668.$

Below is the python script that was written to perform the minor calculations:

from scipy import misc

prob_one_woman = (misc.comb(20, 1)*misc.comb(80, 8))/misc.comb(100,9)
print "Probability this occurs by chance: " +  str(prob_one_woman)
p_value = (misc.comb(20, 1)*misc.comb(80, 8))/misc.comb(100,9) + \
(misc.comb(20, 0)*misc.comb(80, 9))/misc.comb(100,9)
print "P-value with which the null hypothesis can be rejected: " +  str(p_value)


Output:

Probability this occurs by chance: 0.304773971521
P-value with which the null hypothesis can be rejected: 0.42668356013


2. A large jelly bean jar has 20% red jelly beans, 30% blue, and 50% yellow. If 6 jelly beans are chosen at random, what is the chance of getting exactly 2 of each color? What is the name of this distribution?

This is a multinomial distribution.

$\displaystyle N = 6, p_r = 0.2, p_b = 0.3, \text{and } p_y = 0.5$

Want to calculate, P, where P:

$\displaystyle P = \text{Prob}(\text{2 red, 2 blue, 2 yellow}| N, p_r, p_b, p_y) = \frac{6!}{2!2!2!}(0.2)^2(0.3)^2(0.5)^2 = 0.081$

3. A small jelly bean jar has 2 red jelly beans, 3 blue, and 5 yellow. If 6 jelly beans are chosen at random, what is the chance of getting exactly 2 of each color? What is the name of this distribution?

This is a Hypergeometrical Distribution

Want to calculate:

$\displaystyle \text{Prob(2 red, 2 blue, 2 yellow)} = \frac{\text{Number of ways to get outcome}}{\text{Total ways of choosing}} = \frac{ {2 \choose 2} {3 \choose 2} {5 \choose 2}}{{10 \choose 6}} \approx 0.1429$

To Think About:

1. Suppose that, in the population, 82% of people are right-handed, 18% left handed; 49% are male, 51% female; and that handedness and sex are independent. Repeatedly draw samples of N=15 individuals, form the contingency table, and apply the chi-square test for significance to get a p-value, exactly as described in the lecture segment. How often is your p-value less than 0.05? If you get an answer that is different from 0.05, why? Try larger values of N until the answer converges to 0.05. (How are you handling zero draws when they occur?)

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