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

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:

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

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:

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