Inside means plausible, not proven

A confidence interval is a set of plausible values for the parameter. To judge a claim about $p$, find where the claimed value falls:

1. Inside the interval. The claimed value is plausible: this sample gives no reason to doubt it.
2. Outside the interval. The claimed value is not supported: this sample provides evidence against it.

A poll of 400 district voters gives a 95% interval of $(0.572, 0.668)$ for the proportion supporting a measure. A campaign claims that half the district supports it, $p = 0.50$. That value sits below the entire interval, so the data provide evidence against the claim; the plausible values are all well above 0.50.

A different poll gives $(0.47, 0.55)$. Now 0.50 is inside, so it is plausible and cannot be ruled out. Notice what has not been shown: 0.49, 0.52, and 0.54 are equally inside and equally plausible. The interval has not selected 0.50; it has merely failed to exclude it.

Many claims are one-sided: "more than half", "at least 30%", "under a quarter". The interval answers these by position:

1. Interval entirely above the threshold: the data support the claim that $p$ exceeds it. $(0.572, 0.668)$ lies entirely above 0.50, so there is evidence that a majority supports the measure.
2. Interval entirely below: evidence that $p$ falls short of the threshold.
3. Interval straddling it: no conclusion about direction. $(0.47, 0.55)$ contains 0.50, so the data are consistent with a majority, with a minority, and with an exact split.

The straddling case is where the two backwards readings appear. "The interval runs above 0.50 at its top end, so there is a majority" ignores the values below 0.50 that are equally plausible. "The interval contains 0.50, so there is no majority" turns a failure to decide into a decision the other way. The correct conclusion is that this sample cannot distinguish the possibilities, which is a real and reportable finding.

Two overreaches ride along with every interval.

An interval containing a value does not prove that value. The interval $(0.47, 0.55)$ contains 0.50 and also contains every other value in that range; a single sample cannot pick one out. The right phrasing is that 0.50 is a plausible value for $p$, not that $p$ equals 0.50 or that the claim is confirmed.

An interval missing a value does not disprove it either. A 95% interval is built by a method that misses the parameter about 5% of the time, so an interval excluding the true value is uncommon and entirely possible. The claim is that the data provide evidence against the value, at the stated level of confidence.

The confidence level travels with that hedge. It is a property of the method across repeated samples: about 95% of intervals built this way capture $p$. It is not the probability that this interval is right, and a wider interval does not mean a better method, only a more cautious claim.

A complete justification reads as one short paragraph with a fixed structure:

1. State the interval and what it estimates. "We are 95% confident that the interval from 0.572 to 0.668 captures the true proportion of all district voters supporting the measure."
2. Locate the claimed value. "The claimed value of 0.50 lies below the entire interval."
3. Conclude about the parameter, in context. "So these data provide evidence against the claim that only half the district supports the measure, and support the claim that more than half do."

Two habits keep the paragraph correct. Name the population every time, since an interval that never mentions who is being described has already lost its referent. And keep the verbs hedged: evidence against, plausible, consistent with, never proves or shows conclusively. The hedge is not a stylistic softener; it is the accurate report of what a sample can establish.