What a 95% confidence interval actually means

Question

From a sample, you compute a 95% confidence interval for a strategy's mean daily return: $[0.2\text{bp}, 1.8\text{bp}]$ . A colleague concludes: "There is a 95% probability that the true mean lies between 0.2bp and 1.8bp."

Is the statement correct? Give the correct interpretation, and explain what object would justify your colleague's sentence.

Accepted Answer

The statement is wrong (as a frequentist claim). In frequentist statistics the true mean \mu is a fixed, unknown number, it is not random, so it is either in the specific interval [0.2, 1.8] or it isn't. No probability attaches to that event other than 0 or 1; we just don't know which. What is random is the interval. The procedure \bar{X} \pm 1.96\,\frac{s}{\sqrt{n}} produces a different interval for every sample. The 95% is a property of the procedure: before the data are drawn, \mathbb{P}\big(	ext{the interval you will construct covers } \mu\big) = 0.95. Correct sentence: "This interval wa…

What a 95% confidence interval actually means

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