Party Affiliation The following data represent political party by age from a random sample of registered Iowa Voters.

. | 17-29 | 30-44 | 45=64 | 65+ | Total |
---|---|---|---|---|---|

Republican | 224 | 340 | 1075 | 561 | 2200 |

Democrat | 184 | 384 | 773 | 459 | 1800 |

Total | 408 | 724 | 1848 | 1020 | 4000 |

(a) Are the events "Republican" and "30-44" independent? Justify your answer.

(b) Are the events "Democrat" and "65+" independent? Justify your answer.

(c) Are the events "17-29" and "45-64" mutually exclusive? Justify your answer.

(d) Are the events "Republican" and "45-64" mutually exclusive? Justify your answer.

#### Solution

(a)

` $$ \begin{aligned} P(\text{Republican})&=\frac{2200}{4000}\\ &= 0.55. \end{aligned} $$ `

` $$ \begin{aligned} P(\text{30-44 age})&=\frac{724}{4000}\\ &= 0.181. \end{aligned} $$ `

` $$ \begin{aligned} P(\text{Republican} \cap \text{30-44 age})&=\frac{340}{4000}\\ &= 0.085. \end{aligned} $$ `

` $$ \begin{aligned} P(\text{Republican})*P(\text{30-44 age})&=0.55 \times 0.181\\ &=0.09955\\ &\neq P(\text{Republican} \cap \text{30-44 age}) \end{aligned} $$ `

The events "Republican" and "30-44" are not independent.

(b)

` $$ \begin{aligned} P(\text{Democrat})&=\frac{1800}{4000}\\ &= 0.45. \end{aligned} $$ `

` $$ \begin{aligned} P(\text{65+ age})&=\frac{1020}{4000}\\ &= 0.255. \end{aligned} $$ `

` $$ \begin{aligned} P(\text{Democrat} \cap \text{65+ age})&=\frac{459}{4000}\\ &= 0.11475. \end{aligned} $$ `

` $$ \begin{aligned} P(\text{Democrat})*P(\text{65+ age})&=0.45 \times 0.255\\ &=0.11475\\ &=P(\text{Democrat} \cap \text{65+ age}) \end{aligned} $$ `

The events "Democrat" and "65+" are independent.

(c) $\text{(17-29) age} \cap \text{(45-64) age}=\phi$, thus the events "17-29" and "45-64" are mutually exclusive.

(d) $\text{(Republican)} \cap \text{(45-64) age}\neq \phi$, the events "Republican" and "45-64" are not mutually exclusive.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators