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.


LOL In a U.S. Cellular survey of 500 smartphone users, subjects are asked if they find abbreviations (such as LOL or BFF) annoying, and each response was recorded as “yes,” “no,” or “not sure.”

Verified step by step guidance

1. Step 1: Recall the four requirements for a binomial distribution: (1) The procedure must have a fixed number of trials, (2) each trial must have only two possible outcomes (success or failure), (3) the trials must be independent, and (4) the probability of success must remain constant for all trials.
2. Step 2: Analyze the given problem. The survey involves 500 smartphone users, which represents a fixed number of trials. This satisfies the first requirement.
3. Step 3: Examine the possible outcomes for each trial. The responses are recorded as 'yes,' 'no,' or 'not sure,' which means there are three possible outcomes. This violates the second requirement, as a binomial distribution requires only two outcomes (e.g., success or failure).
4. Step 4: Since the second requirement is not satisfied, the procedure does not result in a binomial distribution. Additionally, the problem does not specify whether the trials are independent or whether the probability of success remains constant, but these are secondary considerations since the second requirement is already violated.
5. Step 5: Conclude that the given procedure does not result in a binomial distribution, and the primary reason is that there are more than two possible outcomes for each trial.

Key Concepts

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It requires two outcomes (success or failure), a fixed number of trials, and constant probability across trials. For example, flipping a coin multiple times and counting the number of heads is a classic binomial scenario.

5% Guideline

The 5% guideline is a rule of thumb used to determine if a distribution can be treated as binomial when the sample size is large. It states that if the probability of success is less than 5% and the sample size is large, the trials can be considered independent, allowing for a binomial approximation. This is particularly useful in simplifying calculations for large datasets.

Requirements for Binomial Distribution

For a distribution to be classified as binomial, it must meet specific criteria: a fixed number of trials, only two possible outcomes per trial, independence of trials, and a constant probability of success. If any of these conditions are not met, the distribution may not be binomial. In the given survey, the presence of three response options (yes, no, not sure) violates the binary outcome requirement.