Question

there are 15,958,866 adults in a region. if a polling organization randomly selects 1235 adults without replacement, are the selections independent or dependent? if the selections are dependent, can they be treated as independent for the purposes of calculations?
are the selections independent or dependent?
a. the selections are dependent, because the selection is done without replacement.
b. the selections are independent, because the sample size is small relative to the population
c. the selections are dependent, because the sample size is not small relative to the population.
d. the selections are independent, because the selection is done without replacement

Explanation:

1. Recall the concept of independent and dependent events in sampling: When sampling without replacement, the outcome of one selection affects the probability of the next (since the population size decreases and the composition changes), so selections are dependent.
2. Now check the sample size relative to population: The population is 15,958,866 and the sample is 1,235. The sample size as a proportion of the population is $\frac{1235}{15958866}\approx0.0077\%$, which is very small (typically, if the sample size is less than 5% of the population, we can treat dependent events as independent for calculations, but first, the initial selection is dependent because it's without replacement).
3. Analyze the options:

• Option A: Correctly states selections are dependent because of no replacement.
• Option B: Selections without replacement are not independent, so this is wrong.
• Option C: The sample size is small relative to the population (as shown by the proportion), so this reason is wrong.
• Option D: Sampling without replacement does not make selections independent, so this is wrong.

Answer:

A. The selections are dependent, because the selection is done without replacement.