A hundred samples from the same population, a hundred intervals, and the handful that miss. Then how to compute one.
The population mean is the dotted line at μ = 50, and it never moves. Each horizontal bar is a 95% interval from a fresh sample of 25. Twenty five are drawn already; press play to build all one hundred from scratch.
Every interval is built the same correct way. Some still miss, and that is not a mistake.
So what does 95% attach to? Not to the interval in front of you, which either contains μ or does not. It attaches to the method: repeated indefinitely, 95% of the intervals it produces cover the true mean. You never learn whether yours is one of them.
Interval for a mean, σ unknown: x̅ ± tn−1 × sn−1√n. In practice your calculator does this: give it x̅, sn−1, n and the confidence level, and read off the two ends.
More data. n in the denominator, under a square root. Four times the sample halves the width.
Less variable data. Smaller s means a smaller margin, though that is rarely yours to choose.
Less confidence. A 90% interval is narrower than a 95% one, because you have accepted missing more often. Width and confidence trade against each other and always will.
1. sn−1 = 6 and n = 25. Find the standard error.
2. An interval runs from 47.9 to 52.9. State the sample mean.
3. A 95% interval for a mean is (47.9, 52.9). Which statement is correct?
4. A 90% interval from the same sample would be:
Using sn−1, the sample standard deviation with n − 1 in it, not sn. Calculators give both and the wrong one loses the mark.
Stating the interval as two numbers in brackets, to three significant figures unless told otherwise.
Interpreting it in context without the probability mistake. Safe wording: “we are 95% confident that the mean lies between 47.9 and 52.9”.
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