Segment 32 Sanmit Narvekar

Segment 32

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?

The probability this would occur by chance is (from a hypergeometric distribution):

$\displaystyle Pr(\text{8 men, 1 woman}) = \frac{\binom{20}{1} \binom{80}{8}}{\binom{100}{9}} = 0.3048$

The pvalue with which the statement can be rejected is by testing the probability of seeing 8 men and 1 women OR all men ("something this extreme or more extreme..."). Even without calculating the value, you can see that we will fail to reject the null hypothesis at any of the standard significance levels (e.g. 5% or below), since we have already crossed the threshold.

$\displaystyle Pr(\text{9 men}) = \frac{\binom{80}{9}}{\binom{100}{9}} = 0.1219$

Thus, the pvalue of our dataset under the null hypothesis is 0.3048+0.1219 = 0.4267. Since this is higher than any of the usual significance levels, we cannot rule out the null hypothesis.

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 the multinomial distribution.

$\displaystyle Pr(\text{2 of each color}) = \frac{N!}{n_r!n_b!n_y!} p_r^{n_r}p_b^{n_b}p_y^{n_y} = \frac{6!}{2!2!2!} (0.2)^2(0.3)^2(0.5)^2 = 0.081$

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 the hypergeometric distribution.

$\displaystyle Pr (\text{2 of each color}) = \frac{\binom{2}{2} \binom{3}{2} \binom{5}{2}}{\binom{10}{6}} = \frac{1}{7}$