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. what are the appropriate hypotheses?

○ ( h_0 ): the sample will contain 210 democrats, 170 republicans, and 120 independents.
( h_a ): the sample will not contain 210 democrats, 170 republicans, and 120 independents.

○ ( h_0 ): the sample will not contain 210 democrats, 170 republicans, and 120 independents.
( h_a ): the sample will contain 210 democrats, 170 republicans, and 120 independents.

○ ( h_0 ): the distribution of political party in the municipality is 42% democrat, 34% republican, and 24% independent.
( h_a ): the distribution of political party in the municipality is not 42% democrat, 34% republican, and 24% independent.

○ ( h_0 ): the distribution of political party in the municipality is not 42% democrat, 34% republican, and 24% independent.
( h_a ): the distribution of political party in the municipality is 42% democrat, 34% republican, and 24% independent

Explanation:

In hypothesis testing for distribution of categorical variables (political party affiliation here), the null hypothesis \(H_0\) assumes the distribution in the population (municipality) follows the known national distribution. The alternative hypothesis \(H_a\) is the opposite. The expected counts (e.g., \(0.42\times500 = 210\) Democrats, \(0.34\times500=170\) Republicans, \(0.24\times500 = 120\) Independents) are just calculations based on the hypothesized distribution, not the hypotheses themselves. Hypotheses are about the population distribution, not sample counts.

Answer:

\(H_0\): The distribution of political party in the municipality is \(42\%\) Democrat, \(34\%\) Republican, and \(24\%\) Independent. \(H_a\): The distribution of political party in the municipality is not \(42\%\) Democrat, \(34\%\) Republican, and \(24\%\) Independent. (So the third option is correct)