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?

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

b. the selections are independent, because the selection is done without replacement.

c. the selections are independent, because the sample size is small relative to the population.

d. the selections are dependent, because the sample size is not small relative to the population.

if the selections are dependent, can they be treated as independent for the purposes of calculations?

a. yes, because the sample size is less than 5% of the population

b. no, because the sample size is greater than 5% of the population.

c. yes, because the sample size is greater than 5% of the population.

d. the selections are independent.

Explanation:

First Question (Independence of Selections)

• When sampling without replacement, each selection affects the next (since the population size decreases), so selections are dependent. Option A correctly states this (selection without replacement causes dependence). Option B is wrong (without replacement means dependent). Option C is wrong (sample size relative to population relates to treating as independent for calculation, not the initial independence). Option D is wrong (dependence is due to no replacement, not sample size relative to population for the first part).

• To check if dependent selections can be treated as independent, we use the 5% rule: if the sample size is less than 5% of the population, we can treat them as independent. Calculate 5% of 15,958,866: \( 0.05\times15,958,866 = 797,943.3 \). The sample size (1235) is less than 797,943.3, so it's less than 5% of the population. Thus, we can treat them as independent (Option A is correct). Option B is wrong (sample size is less than 5%). Option C is wrong (reasoning is inverted). Option D is wrong (we already established selections are dependent).

Answer:

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

Second Question (Treating Dependent Selections as Independent)