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

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− | <b>To Calculate </b> | + | <b>To Calculate: </b> |

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? | 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? | ||

+ | |||

+ | <math>\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. </math> | ||

+ | |||

+ | 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: | ||

+ | |||

+ | <math>\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. </math> | ||

+ | |||

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

+ | |||

+ | <pre> | ||

+ | 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) | ||

+ | </pre> | ||

+ | |||

+ | <b> Output: </b> | ||

+ | <pre> | ||

+ | Probability this occurs by chance: 0.304773971521 | ||

+ | P-value with which the null hypothesis can be rejected: 0.42668356013 | ||

+ | </pre> | ||

+ | |||

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? | 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? | ||

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Want to calculate: | Want to calculate: | ||

− | <math> \text{Prob(2 red, 2 blue, 2 yellow)} =\frac{ {2 \choose 2} {3 \choose 2} {5 \choose 2}}{{10 \choose 6}} </math> | + | <math> \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 </math> |

− | <b>To Think About</b> | + | <b>To Think About:</b> |

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?) | 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?) | ||

<b> Back To: </b> [[Eleisha Jackson]] | <b> Back To: </b> [[Eleisha Jackson]] |

## Latest revision as of 12:13, 13 April 2014

**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?

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = 6, p_r = 0.2, p_b = 0.3, \text{and } p_y = 0.5 }**

Want to calculate, P, where P:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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?)

** Back To: ** Eleisha Jackson