Questions: Which of the following best describes the "5% guideline for cumbersome calculations"? Choose the correct answer below. A. Whenever probabilities are very difficult to calculate for multiple events, those events can be treated as being independent. B. When a sample is no more than 5% of a population, items selected without replacement should be treated as dependent events. C. A sample should always be at least 5% of a population. D. When a sample is no more than 5% of a population, items selected without replacement can be treated as independent events. E. When a sample is more than 5% of a population, items selected without replacement can be treated as independent events.

Which of the following best describes the "5% guideline for cumbersome calculations"?

Choose the correct answer below. A. Whenever probabilities are very difficult to calculate for multiple events, those events can be treated as being independent. B. When a sample is no more than 5% of a population, items selected without replacement should be treated as dependent events. C. A sample should always be at least 5% of a population. D. When a sample is no more than 5% of a population, items selected without replacement can be treated as independent events. E. When a sample is more than 5% of a population, items selected without replacement can be treated as independent events.

Transcript text: Which of the following best describes the " $5 \%$ guideline for cumbersome calculations"? Choose the correct answer below. A. Whenever probabilities are very difficult to calculate for multiple events, those events can be treated as being independent. B. When a sample is no more than $5 \%$ of a population, items selected without replacement should be treated as dependent events. C. A sample should always be at least $5 \%$ of a population. D. When a sample is no more than $5 \%$ of a population, items selected without replacement can be treated as independent events. E. When a sample is more than $5 \%$ of a population, items selected without replacement can be treated as independent events.

Solution

Solution Steps

To solve this problem, we need to understand the concept of the "$5\%$ guideline for cumbersome calculations." This guideline is used in statistics to simplify calculations when dealing with probabilities. The correct description of this guideline is that when a sample is no more than $5\%$ of a population, items selected without replacement can be treated as independent events. This is because the impact of removing a small number of items from a large population is negligible, allowing us to assume independence.

Step 1: Understanding the 5% Guideline

The "$5\%$ guideline" is a statistical principle that helps simplify calculations involving probabilities when sampling from a population. It states that if a sample size is no more than $5\%$ of the total population, the items selected without replacement can be treated as independent events. This is because the removal of a small number of items from a large population does not significantly affect the probabilities of subsequent selections.

Step 2: Analyzing the Options

We need to evaluate the provided options to identify which one accurately describes the "$5\%$ guideline." The options are:

A: Incorrect, as it does not relate to the $5\%$ guideline.
B: Incorrect, as it contradicts the guideline.
C: Incorrect, as it misstates the requirement for sample size.
D: Correct, as it accurately reflects the guideline.
E: Incorrect, as it misapplies the concept of independence.

Step 3: Selecting the Correct Answer

Based on the analysis, the correct description of the "$5\%$ guideline" is option D: "When a sample is no more than $5\%$ of a population, items selected without replacement can be treated as independent events."

Final Answer

$\boxed{D}$

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