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Here is the problem: A sample of 100 has mean 72; the population standard deviation is 12. Find the 99 percent confidence interval for the mean.

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Build a 99 percent confidence interval for the population mean μ.

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Standard error of the mean.

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Critical value for 99 percent confidence. The middle 99 percent leaves 0.5 percent in each tail. σ is treated as known, so the standard normal distribution gives the critical value.

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Margin of error.

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Interval: estimate plus or minus margin.

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So the answer is 99 percent CI: (68.909, 75.091); margin of error 3.091.

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We are 99 percent confident that the population mean μ lies between 68.909 and 75.091. The confidence describes the method: about 99 percent of intervals built this way from repeated samples would capture the true value.

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The interval is centred on the estimate: (68.909 plus 75.091) / 2 equals 72.