Question 6(Multiple Choice Worth 5 points) (04.05 MC) In a recent text message survey, 4,000 randomly selected teenagers were asked to cite their favorite Florida college football team. Twenty-wo of 40 teenagers said the Florida Gators are their favorite team. A 99\% confidence interval to estimate the true proportion of teenagers who like the Florida Gators is found to be ( \( 0.3474,0.7526 \) ), Which of the following is a correct interpretation of the confidence tevel? Ninety-nine percent of all samples of this size would yield a confidence interval of \( (0.3474,0.7526) \). Ninety-nine percent of all the samples of size 4,000 lie in the confidence interval \( (0.3474,0.7526) \). Ninety-nine percent of the time, the procedure used to generate this interval will capture the true proportion of teenagers who like the Florida Gators. There is a \( 99 \% \) chance that randomly selected teenagers will be part of the \( 55 \% \) who like the Florida Gators.
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Confidence intervals can seem tricky, but here's a fun way to think about it! When we say there’s a 99% confidence level on that interval of \( (0.3474,0.7526) \), we could picture it like a game where we throw darts at a target of true proportions. Out of 100 similar dart-throwing sessions, 99 times, our throws would land within that interval, hitting the mark just right! Now, let’s address a common misunderstanding: it’s easy to mix up probability with certainty. The correct interpretation means that if we were to repeat our survey 100 times, we’d expect 99 of those confidence intervals to include the true proportion of teenagers who like the Gators. It’s a reliable method, but it doesn’t guarantee that this single interval we have captures the true proportion!
