A radio talk show host with a large audience is interested in the proportion \( p \) of adults in his listening area who think the drinking age should be lowered to eighteen. To find this out, he poses the following question to his listeners:

"Do you think that the drinking age should be reduced to eighteen in light of the fact that eighteen-year-olds are eligible for military service?"

He asks listeners to phone in and vote "Yes" if they agree the drinking age should be lowered and "No" if not. Of the 100 people who phoned in, 70 answered "Yes."

Which of the following conditions for inference about a proportion using a confidence interval are violated?

1. The data are an SRS (Simple Random Sample) from the population of interest.
2. The population is at least 10 times as large as the sample.
3. \( n \) is so large that both the count of successes \( np \) and the count of failures \( n(1 - p) \) are 10 or more.
4. There appear to be no violations.
5. More than one condition is violated.

The conditions likely violated are the need for a simple random sample and the potential for response bias due to question phrasing, while the population size and success-failure count conditions are probably met.

Explanation:

The conditions for inference about a proportion using a confidence interval are likely violated in this scenario. First, the sample does not appear to be a simple random sample (SRS) from the population of interest since it only includes listeners who chose to call in, potentially leading to voluntary response bias.

Second, the way the question is phrased could lead to response bias because it provides a justification for lowering the drinking age before asking for the listener's opinion, which may influence the results. Lastly, assuming there is a large population of listeners, conditions 2 and 3 regarding the population size and success-failure counts are probably met.

The condition violated is number 3) n is so large that both the count of successes np and the count of failures n(1 - p) are 10 or more. In this case, the host asked 100 people to phone in and vote, and 70 answered 'Yes.' Since the sample size is only 100, the count of successes (70) does not meet the requirement of being at least 10 or more. Therefore, the condition is violated.