Zero is the no-change value

A 95% interval for the mean battery lifetime is $(120.8, 128.4)$ hours, and the manufacturer claims a mean of 128 hours. That value sits inside the interval, so these data do not contradict the claim: 128 hours is a plausible mean, along with 122, 125, and every other value in the range.

Notice what has not been shown. The sample mean was 124.6, below the claim, and the interval still contains 128. A point estimate landing below a claimed value is not evidence against it until the claimed value falls outside the interval entirely.

Had the interval come out at $(118.2, 125.4)$, the claim of 128 would lie above the whole interval, and the data would give evidence against it. That is the only situation that produces a verdict against a claimed value.

When the parameter is a mean difference $\mu_d$, the value 0 says the mean did not change. Three positions:

1. Entirely above 0: evidence of an increase, in the stated order of subtraction.
2. Entirely below 0: evidence of a decrease.
3. Containing 0: no convincing evidence of any change, since increases, decreases, and no change are all plausible.

Twelve students take a pre-test and a post-test; the mean improvement is 3.6 points with $s_d = 5.4$, giving $SE = 1.559$, $df = 11$, $t^{*} = 2.201$, and an interval of $3.6 \pm 3.43$, or about $(0.17, 7.03)$. It lies entirely above 0, so there is evidence of a mean improvement, of somewhere between about 0.2 and 7.0 points.

Had the interval been $(-1.2, 4.8)$, the correct sentence would be that there is no convincing evidence of a change. It would not be that the program had no effect: 4.8 points is inside that interval, and so is 0, and one sample of 12 cannot separate them. And the sign of the endpoints is unreadable without the order of subtraction, so "after minus before" belongs in the report.

The margin of error is $t^{*}\frac{s}{\sqrt{n}}$, and every one of its pieces is about random sampling:

1. Confidence level, through $t^{*}$. Higher confidence widens the interval.
2. Variability, through $s$. More variable data widens it.
3. Sample size, through $\sqrt{n}$. Quadrupling $n$ halves the margin; doubling $n$ divides it by only about 1.41.

What it does not cover is everything else that can go wrong: a leading question, nonresponse, a miscalibrated instrument, a sample drawn from the wrong population. Those shift the whole sampling distribution, and no arithmetic in the formula detects them. A survey with a 3-point margin of error and a badly worded question is wrong by far more than 3 points, and the margin of error will never say so.

One more misread: narrowing an interval by lowering the confidence level is not free precision. The interval shrinks because it now claims less often to have captured the parameter.

Four parts:

1. State: "We are 95% confident the interval from 0.17 to 7.03 points captures the true mean improvement, post minus pre, for students in this program."
2. Locate the claimed value: "The value 0 lies below the entire interval."
3. Conclude: "So there is convincing evidence of a mean improvement."
4. Give the magnitude: "of roughly 0.2 to 7.0 points."

Keep the verbs hedged: convincing evidence, plausible, never proves. And keep the causal claim out unless the design supports it. Students who took the program were not randomly assigned to it in most such studies, and even in a before-and-after design with no control group, ordinary practice effects and maturation are alternative explanations that the interval cannot rule out.