Question

Sample for a Poll There are 15,524,971 adults in Florida. If The Gallup organization randomly selects 1068 adults without replacement, are the selections independent or dependent? If the selections are dependent, can they be treated as being independent for the purposes of calculations?

Short answer

The selections are dependent as they have been made without replacement.

Since the sample size is less than 5% of the population size, the selections can be treated as independent for calculations.

Step-by-step solution

Given information

Out of 15,524,971 adults of Florida, 1068 are selected randomly without replacement.

Define the sampling methods

There are two main sampling methods:

• Sampling with replacement: In this method, each unit selected from the sample is returned to the sample for the subsequent draw. Here,events are independent of each other. This means that the outcome of an event is not affected by the outcome of the preceding event.
• Sampling without replacement: In this method, each unit selected from the sample is not returned to the sample for any subsequent draw. Here,events are dependent on each other. This means that the outcome of an event is affected by the outcome of the preceding event

The events can be considered to be independent even if the sampling method is without replacement when the sample size is less than or equal to 5% of the population size.

The proportion of samples from the population

Here, the selections of adults are made without replacement. Thus, the selections are dependent.

Let n be the sample size and N be the population size.

$$\begin{gathered} \text{Sample}\text{size}\text{proportion}=\frac{n}{N}\times 100 \\ =\frac{1068}{15524971}\times 100 \\ =0.0068\% \end{gathered}$$

As the percentage equal to 0.0068% is less than 5%, the selections can be considered independent for calculations.