In Exercises 5–12, determine whether the given procedure results in a binomial distribution or a distribution that can be treated as binomial (by applying the 5% guideline for cumbersome calculations). For those that are not binomial and cannot be treated as binomial, identify at least one requirement that is not satisfied.


In a Pew Research Center survey, 3930 subjects were asked if they have ever fired a gun, and the responses consist of “yes” or “no.”

Verified step by step guidance

1. Step 1: Recall the criteria for a binomial distribution. A binomial distribution requires the following conditions: (1) The experiment consists of a fixed number of trials, n. (2) Each trial has only two possible outcomes (success or failure). (3) The probability of success, p, remains constant for each trial. (4) The trials are independent of each other.
2. Step 2: Analyze the given problem. The survey involves 3930 subjects being asked a question with two possible responses: 'yes' or 'no.' This satisfies the condition of two possible outcomes (success or failure).
3. Step 3: Check if the trials are independent. In this case, the responses of the subjects are likely independent, as one person's response does not influence another's.
4. Step 4: Verify if the probability of success (e.g., answering 'yes') remains constant. Since the survey is conducted on a large population, the probability of success can be assumed to remain constant for each subject.
5. Step 5: Apply the 5% guideline for cumbersome calculations. If the sample size (3930) is less than 5% of the population size, the trials can be treated as independent even if the population is finite. If this condition is met, the procedure results in a binomial distribution or can be treated as binomial.

Key Concepts

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two outcomes (success or failure) and requires a set number of trials, a constant probability of success, and independence between trials.

5% Guideline

The 5% guideline is a rule of thumb used to determine whether a sample can be treated as binomial when sampling without replacement. If the sample size is less than 5% of the population, the trials can be considered independent, allowing the binomial distribution to be applied even when sampling from a finite population.

Requirements for Binomial Distribution

For a distribution to be classified as binomial, it must meet specific criteria: a fixed number of trials, two possible outcomes for each trial, a constant probability of success, and independence of trials. If any of these conditions are not met, the distribution may not be binomial, and identifying the unmet requirement is crucial for proper analysis.