Question

Leanne manages an apartment building that houses 100 residents. she read an article claiming that 15 % of people in their state were senior citizens, and she thought that it may be higher for residents in her building. she took a random sample of 20 residents, and she wants to test h₀ : p = 0.15 versus h ₐ : p > 0.15 , where p is the proportion of residents that are senior citizens. which conditions for performing this type of test did leanne's sample meet?

O the data is a random sample from the population of interest
O b the expected counts of successes and failures are both sufficiently large
O individual observations can be considered independent.

Answer 1

Sample random and observations independent, but expected successes too low for a reliable chi-square test. Consider alternatives like Fisher's exact test. So, A and B conditions are met.

Leanne's sample met two out of the three conditions for performing a chi-square test to compare the proportion of senior citizens in her building to the state average:

Conditions met:

A. Random sample: The prompt states that Leanne took a random sample of 20 residents, which satisfies this condition.

C. Independent observations: It is reasonable to assume that the observations are independent, meaning the status of one resident as a senior citizen does not influence the status of another. This is typically true for residents in an apartment building.

Condition not met:

B. Expected counts: While the sample size is 20, which is generally considered adequate, the expected number of successes (residents being senior citizens) under the null hypothesis (15% * 20 = 3) is not sufficiently large. The rule of thumb for chi-square tests is that both the expected number of successes and failures should be greater than 10. In this case, the expected number of failures (17) is sufficient, but the expected number of successes is not.

Therefore, while Leanne's sample meets two important conditions, the low expected number of successes under the null hypothesis makes the chi-square test less reliable in this case. She should consider using a different test, such as Fisher's exact test, which is more suitable for small samples with low expected counts.