LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A 95 percent confidence interval is a random interval that, under repeated samples, covers the fixed population mean about 95 times out of 100. It is not a 95 percent chance that the mean sits between the two numbers you just computed. In the lecture’s numerical sketch a mean of 50, from 100 observations and a known spread of 10, produces a 95 percent interval running about from 48 to 52. Those ends are one interval, not a cage around a random mean.
Why it matters
The wrong sentence is common and it sounds like the right one. “There is a 95 percent chance the true mean is between 48 and 52” treats the mean as the random object and the ends as fixed. The construction does the opposite. The mean is a property of the population. The ends move because the sample mean moves. Once you have printed 48 and 52, that particular pair either contains the mean or it does not. The 95 percent is a property of the procedure that prints pairs, judged across repetitions.
The idea in plain words
The sketch assumes the population standard deviation is known and equal to 10. If that spread had been estimated from the sample, the lecture says the cutoff would come from a t distribution instead. With a known spread, a 95 percent interval leaves 2.5 percent in each tail. The lecture takes the 0.975 point of the standard normal, via the percent-point function in SciPy, as that cutoff. Plug in the sample mean 50, the size 100, and the standard deviation 10, and the ends come out near 48 and 52.
Now imagine the population is people’s heights and you do not know the mean height. One sample of 100 people produces one interval. A second sample of 100 different people produces another interval, with different ends, because the sample average changed. Do that 100 times and you have 100 intervals aimed at the same unknown mean. The lecture’s chart draws those intervals against a horizontal line at the true mean. Most segments cross the line. A few miss it entirely. The misses are the procedure failing on those samples. Roughly 95 of the 100 segments are supposed to be the ones that hit, which is what “95 percent confidence” is counting. It is not counting a probability inside any one segment after the segment is drawn.
The ends depend on the sample average and the sample size, and, in this sketch, on a population standard deviation that is treated as fixed. You are not sampling from “between 48 and 52” as if that were a known bin of heights. You are sampling from the population, building an interval, and only afterward noticing whether the bin you built happened to cover the mean.
How to use it
When you report an interval, name the procedure and the repetition. A sentence that matches the lecture is: if we drew samples of this size again and again and built the interval the same way, about 95 percent of those intervals would contain the population mean. Then give the numerical ends as the interval this sample produced. Do not attach a probability to the mean being inside these exact ends.
What usually goes wrong
The business-school sentence the lecture is annoyed by is the “95 percent chance the mean is in here” sentence. It is wrong for a completed interval because the mean is not the random piece. Repeating it with a footnote does not fix it. The repetition has to be in the sentence: many intervals, one mean, about 95 percent coverage.
Questions people ask
What is wrong with saying the mean is probably between 48 and 52?
Those ends came from one sample. The mean is either inside them or not. The 95 percent describes how often the method’s intervals cover the mean across repeated samples, not a probability glued to this pair of ends.
Why might the next interval have different ends?
Because the next sample has a different average. The lecture’s height picture draws 100 intervals from 100 samples of 100 people, and the segments do not all sit on top of each other. A few miss the true mean entirely.
When would this sketch use a t cutoff instead?
When the standard deviation is estimated from the sample rather than treated as a known population value of 10. The lecture’s 48-to-52 sketch is the known-spread case.