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?

A. No, the random condition is not met.
B. No, the 10% condition is not met.
C. No, the Large Counts condition is not met.
D. Yes, all of the conditions for inference are met.

Answer 1

All conditions for inference are met: the random condition is satisfied as a random sample was taken, the 10% condition is likely satisfied as the sample size is unlikely to exceed 10% of the municipality's population, and the Large Counts condition is satisfied as the expected number of successes and failures for each political affiliation in the sample is greater than 10.

Answering your question on whether the conditions for inference are met to know if the distribution of political party affiliation among citizens in a local municipality differs from the nationwide percentages, we need to check three main conditions:

The random condition requires that the sample must be random. Since the question states that a random sample was selected, this condition is met.

• The 10% condition states that the sample size must not be more than 10% of the population. This is typically met in real-world settings because most populations are large, and a 500-person sample is unlikely to be more than 10% of a municipality's population.
• The Large Counts condition requires that the expected number of successes and failures must both be at least 10. This can be checked by calculating the expected numbers: for Democrats 42% of 500 is 210, for Republicans 34% of 500 is 170, and for Independents 24% of 500 is 120. Since all expected numbers are greater than 10, this condition is met.

Given that all conditions are met, the correct answer is:

D. Yes, all of the conditions for inference are met.

Answer 2

D. Yes, all of the conditions for inference are met.

Step 1

Introduction:

The population percentages of political affiliations are 42% Democrats, 34% Republicans, and 24% Independents.

Step 2

Step-by-step explanation:

It is given that a random sample of 500 citizens of the municipality is considered. Thus, the random condition is met.

The sample size must not exceed 10% of the population size, that is, the population size must be more than 10 times the sample size. Now, if 500 were 10% of the total population of the municipality, the population size for the municipality would be 500 × 10 = 5000. It is logical to assume that a municipality has more than 5000 people. Thus, the 10% condition seems to be satisfied.

The sample size must be large enough, which is usually taken to be at least 30. The sample size of 500 is surely much larger compared to that. Thus, the large counts condition is met.

The correct option is, "yes, all of the conditions for inference are met.