Question

according to a government website, 42% of us citizens are democrats, 34% are republicans, and 24% are independents. a local municipality would like to know if the distribution of political party affiliation among its citizens differs from the nationwide percentages. a random sample of 500 citizens of the municipality is selected. are the conditions for inference met?
no, the random condition is not met.
no, the 10% condition is not met.
no, the large counts condition is not met.
yes, all of the conditions for inference are met.

Explanation:

Step1: Check Random Condition

The problem states a random sample of 500 citizens is selected. So the random condition is met.

Step2: Check 10% Condition

We assume the municipality has more than \(500\times10 = 5000\) citizens (a reasonable assumption for a municipality). So the 10% condition (sample size \(n\) is less than 10% of the population size \(N\), \(n<0.1N\)) is met.

Step3: Check Large Counts Condition

For Democrats: \(n\times p_{Dem}=500\times0.42 = 210\geq10\)
For Republicans: \(n\times p_{Rep}=500\times0.34=170\geq 10\)
For Independents: \(n\times p_{Ind}=500\times0.24 = 120\geq10\)

Answer:

Yes, all of the conditions for inference are met.