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 dependent, because the sample size is not small relative to the population.
c. the selections are independent, because the sample size is small relative to the population.
d. the selections are independent, because the selection is done without replacement.

Explanation:

Step1: Recall dependence - independence rule

Selections without replacement are generally dependent.

Step2: Check the 5% guideline for treating as independent

The sample size $n = 1235$ and the population size $N=15958866$. Calculate the ratio $\frac{n}{N}=\frac{1235}{15958866}\approx0.0000774$. Since $\frac{n}{N}<0.05$ (the 5 - percent guideline), although the selections are made without replacement (so they are technically dependent), they can be treated as independent for the purposes of calculations.

Answer:

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