Exercise 3: Correcting a Misinterpreted Confidence Interval — Possible Solution ==================================================================== WHAT'S WRONG WITH THE TEAMMATE'S RESTATEMENT ------------------------------ The teammate says "there's a 95% chance the real value is in that range" - treating the 95% as a probability attached to this one specific, already-computed interval (190ms to 210ms). Per this chapter's own explanation, that's not what a confidence interval means. The true population mean is a single, fixed (though unknown) number - it either already falls inside [190, 210] or it doesn't, with no randomness left to assign a probability to once the interval has been computed. The randomness in this whole process belongs to the SAMPLING and the INTERVAL-BUILDING procedure, not to the true mean itself. THE STATISTICALLY CORRECT VERSION ------------------------------ Per this chapter's own frequentist interpretation: if this exact sampling-and-interval-building process were repeated many times - drawing a new sample each time and building a new 95% confidence interval from it - approximately 95% of those resulting intervals would contain the true population mean. The 95% is a statement about how reliable the METHOD is across many repeated uses, not a probability statement about this one particular interval, [190, 210], containing the truth. A CONCRETE WAY TO SEE THE DIFFERENCE ------------------------------ Once this specific sample has been collected and this specific interval has been computed, that interval is fixed - it either contains the true mean or it doesn't, with certainty, even though we don't know which case we're in. The 95% confidence describes the long-run success rate of the recipe used to build it, the same way saying "this coin-flipping process lands heads 50% of the time" doesn't mean any specific already-flipped coin currently sitting heads -up on the table has a 50% chance of being heads. WHY THIS WORKS AS AN ANSWER ------------------------------ The correction is grounded directly in this chapter's own stated distinction between a probability about a fixed true value and a statement about a method's long-run reliability, and restates the teammate's claim in the chapter's own correct frequentist language rather than simply asserting the original phrasing is wrong.