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Determine whether the given procedure results in a binomial distribution. If it is not binomial, identify the requirements that are not satisfied. Randomly selecting 50 adults and asking if they like rap music, recording Yes or No Choose the correct answer below. A. No, because there are more than two possible outcomes and the trials are not independent. B. Yes, because all 4 requirements are satisfied. Co, because there are more than two possible outcomes. Do, because the probability of success does not remain the same in all trials.

Ask by Wright Bird. in the United States
Feb 13,2025

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Answer

Yes, the procedure results in a binomial distribution because it meets all four requirements: a fixed number of trials, independent trials, two possible outcomes, and a constant probability of success.

Solution

The scenario is as follows: we randomly select 50 adults, ask each one if they like rap music, and record their answer as either "Yes" (success) or "No" (failure). For a distribution to be binomial, it must satisfy these four conditions: 1. A fixed number of trials (n). 2. Each trial is independent. 3. Each trial results in one of exactly two outcomes (commonly called "success" and "failure"). 4. The probability of success (and thus failure) remains constant for each trial. Let’s check each requirement: 1. Fixed Number of Trials: We are selecting 50 adults, so n = 50, which is fixed. 2. Independence: Assuming the adults are randomly selected from a large population, one person’s response does not affect another’s, so the trials can be taken as independent. 3. Two Possible Outcomes: The answer is either "Yes" or "No". Even though opinions are being recorded, the possible responses are reduced to two outcomes for our count (liking rap music or not). 4. Constant Probability: Since the adults are randomly selected and assuming that the population is large, the probability that an adult likes rap music is the same for each individual. Since all four requirements are met, the experiment follows a binomial distribution. Therefore, the correct answer is: B. Yes, because all 4 requirements are satisfied.

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The correct answer is D. It's not a binomial distribution because the probability of success does not remain the same in all trials. In a binomial distribution, the probability of success must remain constant for each trial, but preferences for music, like rap, can vary based on external influences such as recent trends or personal experiences, leading to changing probabilities. This procedure also fails to meet the "fixed number of trials" condition properly. In theory, asking 50 adults creates a fixed number of trials, but if the sampling method introduced biases or dependencies in responses (like asking friends or family), it could affect the independence of trials and compromise the validity of the distribution.

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