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?

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

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.

$N=6,p_{r}=0.2,p_{b}=0.3,{\text{and }}p_{y}=0.5$

Want to calculate, P, where P:

$P={\text{Prob}}({\text{2 red, 2 blue, 2 yellow}}|N,p_{r},p_{b},p_{y})={\frac {6!}{2!2!2!}}(0.2)^{2}(0.3)^{2}(0.5)^{2}=0.081$

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:

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

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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